Hovercraft

Hovercraft is a build event with a physics core. Before the tournament, each team designs, builds and calibrates a self-propelled vehicle that floats on a cushion of air, and for 2027 the vehicle also has to carry nickels supplied by the Event Supervisor down a timed 185 cm track. Three scores make up each run. The Distance Score rewards reaching the finish line, the Time Score rewards finishing as close as possible to a Target Time (TT) announced at the start of your competition time, and the new Mass Score rewards the number of nickels the vehicle carries the whole way. Speed alone wins nothing. A vehicle that repeats a known average speed with a known load beats one that is fast but unpredictable. Strong teams understand how a fan keeps a loaded craft off the floor, why a heavier craft needs more cushion pressure and a longer launch, how to hold a payload without tape, and why the score formula punishes an early finish exactly as much as a late one.

The first half of this guide is the build: rules, payload, vehicle physics, scoring, design, calibration and strategy. The second half, starting at Units, vectors and measurement, is a full course in the mechanics and fluid physics that the event's archived written tests ask about, from kinematics and Newton's laws through momentum, energy, friction, drag, pressure, buoyancy and Bernoulli, ending with how real hovercraft work and their history. The same physics is what you need to explain your own vehicle to the Event Supervisor. A condensed formula sheet is at the end in Condensed facts.

Event format

Impound changes how you prepare. Anything you want during competition time, including spare parts and your calibration sheets, must go into impound with the vehicle. Nothing new on paper may come into the competition area afterwards. Plan your calibration tables and your run log before you check in.

What the rules cover

The 2027 rules manual has the same Hovercraft text for Division B (pages B31 to B34) and Division C (pages C37 to C40) except for one number: the top of the Target Time range. It has seven numbered parts. The table below summarizes each part in plain words; read the manual itself for exact wording.

If you are working from an older guide or last season's notes, three changes matter most. First, the bonus run and Bonus Score are gone; the Final Score is simply your best Run Score. Second, the Distance Score and Time Score are now worth 40 points each, and the remaining 20 points come from the nickels. Third, competition time ends early once you have made two complete runs or two incomplete runs, so you get very few attempts.

The 2027 payload: ES-provided nickels

The Event Supervisor brings a fixed supply of U.S. nickels for every team:

A U.S. nickel has a specified mass of 5.000 g, a diameter of 21.21 mm and a thickness of 1.95 mm, so a full roll is a cylinder roughly 21 mm across and about 80 mm long once wrapped. Build your payload holder around those dimensions, but leave a little slack: bank wrappers vary, and the half roll is shorter and may be slightly irregular where the ends were trimmed.

The rules on carrying them are strict, and most of them exist so that every team works with the same unmodified load:

• You choose how much to carry, from nothing up to the full supply, and you may change the load from one run to the next.
• You may not unroll, break or otherwise alter the packaging of the rolls.
• You may not use tape or adhesive to fix rolls or nickels to the vehicle. A vehicle that needs tape or glue to hold the nickels may not carry any masses for the rest of its competition time.
• The nickel load goes on the vehicle before the run begins, when you place the craft behind the start line.
• Any roll or nickel that falls off before the Event Supervisor tells you to switch off at the end of the run does not count toward the Mass Score. Anything falling off (a vehicle part, a roll or a single nickel) also costs one construction penalty for that run, not one per item.

The practical consequence is that the holder is part of your vehicle design. You need a tray, cradle or set of slots that grips each roll and keeps loose coins from sliding out, using only its own shape, clips or covers. Because the holder must work without tape, test it with real bank rolls, and test it with a hard stop as well as a smooth run.

Vehicle physics: floating on an air cushion

A legal craft must levitate on a cushion of air while it moves down the track. The Event Supervisor may test this by pressing the craft down slightly: if it rises again, it is levitating. For 2027 the rules also say that touching the track surface and the inner faces of the side rails is allowed, while touching the tops or outer faces of the rails is not. So the goal is not zero contact. It is to keep enough air under the craft that contact friction stays small, and to keep the craft level and stable while it carries its load.

Here is the causal chain that holds a craft up:

↑ fan flow → ↑ pressure in the plenum (the space under the deck) → ↑ net upward force on the deck → the craft rises → the skirt gap opens → more air escapes under the skirt → the plenum pressure falls back until the lift force equals the weight.

The craft settles at the height where the air the fan supplies equals the air leaking out under the skirt. That equilibrium is why a skirt with a larger gap needs more fan flow, and why a skirt that is too tight makes the craft rub. The plenum is the enclosed region under the vehicle that the fan fills; it is the single word that connects the fan to the cushion.

Two design consequences follow. The first is the footprint. The craft must fit inside a 40.0 cm cube when it is levitated and ready to run, so the cushion can never be larger than 40 cm by 40 cm, which is 0.160 m². A larger cushion lowers the pressure needed for a given mass. The second is mass. With up to a kilogram of nickels available, the payload can easily outweigh the craft itself, so a light structure leaves more of the fan's capacity for the load.

The speed of air escaping under the skirt is set by the pressure difference. For an ideal, loss-free gap, Bernoulli's principle gives $v = \sqrt{2\,\Delta p / \rho}$, with $\rho \approx 1.2\ \text{kg/m}^3$ for air near sea level. Real leakage is slower because the flow contracts and loses energy at the gap, so treat the formula as an upper bound that shows the scaling.

How much harder the fan works with a full load

The ideal power the lift fan delivers to the cushion is the pressure times the volume of air per second that leaks out. Both rise when you add nickels: the pressure rises in proportion to the mass, and the leak speed rises with the square root of the pressure.

Where to put the load

The cushion pushes up over its whole area, so the craft balances best when its center of mass sits over the center of the cushion. If the nickels sit toward one side, that side sinks, its skirt gap closes and it drags on the track, while the opposite side lifts and leaks more air. The result is a craft that tilts, rubs and yaws toward the heavy side. Mount the payload holder near the middle of the deck, and if you plan to carry partial loads, arrange the slots so that every load you might choose stays balanced: for example, fill rolls in symmetric pairs, and place the half roll and loose nickels on the centerline.

Thrust, drag and motion

Many hovercraft use two air systems. A lift fan feeds the cushion and one or more propellers push the craft forward. Others use a single fan and split its flow between the cushion and a rear thrust duct. Either way, the 2027 rules require all lift and all thrust to come from air pressure. Wheels that drive the craft, rubber bands or springs that push it, and any other non-air propulsion are out. Non-electric energy may still power other functions, such as a timing mechanism.

Thrust works through Newton's third law. A propeller pushes air backward, and the air pushes the craft forward with an equal and opposite force. The craft's acceleration then follows Newton's second law.

A steady speed means zero net force, so drag equals thrust. Students who believe a moving object needs a continuing net force to keep moving make the classic error here. A net force changes velocity; it is not needed to hold a velocity once thrust balances drag and friction.

Average speed is what the Target Time measures

The run time runs from the moment the dowel crosses the start line until it crosses the finish line, 185.0 cm later. The Target Time is therefore a target for the average speed over the timed section:

$$\bar v = \frac{d}{t} = \frac{1.85\ \text{m}}{TT}$$

At the extremes, a 6.0 s target needs 1.85 / 6.0 = 0.308 m/s. The Division B maximum of 18.0 s needs only 0.103 m/s, and the Division C maximum of 26.0 s needs 0.071 m/s. A 10.0 s target needs 0.185 m/s. The slow end is hard in its own way: at 0.07 m/s, small amounts of friction or a slight slope in the track can stall the craft, and a stall of 3 seconds makes the run incomplete.

Using the start position: a running start

For 2027 you may place the craft anywhere behind the start line, held by the Event Supervisor's wood block. Because timing begins only when the dowel crosses the line, the launch acceleration can happen before the clock starts. If the craft has already reached its cruising speed when the dowel crosses, the timed section is run at nearly constant speed and $t \approx 1.85 / v$, which is far easier to calibrate.

There is a limit. If the dowel does not cross the start line within 3 seconds of launch, that attempt is not counted as a run. So you need to know how far back you can start.

Compare a standing start with the dowel on the line. The craft would spend its first 1.57 s at below-cruise speed, adding $v / (2a) = 0.185 / 0.236 = 0.78$ s to the run. That is 3.1 Time Score points lost at a 10.0 s target if you did not plan for it, and the loss changes with the load.

Kinetic energy, friction and drag at low speed

Air drag is far smaller at these speeds. With the drag equation $F_d = \tfrac{1}{2}\rho v^2 C_d A$, a rough $C_d \approx 1$ for a blunt shape, a frontal area of 0.040 m² and $v = 0.185$ m/s, $F_d \approx 0.5 \times 1.2 \times 0.185^2 \times 1 \times 0.040 \approx 0.0008$ N. Contact friction is the force that decides whether your run is repeatable, and it grows when a heavy load is badly placed or the cushion is underpowered. That is why the base surface rules matter: the track may sit on a table, a built-in base or the floor, small imperfections are allowed, and you may only dry-sweep it, not wet it or treat it.

Impulse, the barrier, and keeping the nickels aboard

The Event Supervisor supplies a cushioned barrier at least 5.0 cm past the finish line. For 2027 the barrier matters to your score, because any nickel that falls off before you are told to switch off does not count and triggers a construction penalty. The stop at the barrier is often the hardest moment for the payload.

The same reasoning applies to rolls. A roll lying lengthwise across the craft can roll forward at the stop, so cradles that hold each roll in a groove or under a retaining bar are safer than a flat deck. A soft bumper on the nose of the craft also lengthens the stopping time and reduces the force on the payload, the dowel and the frame.

Construction rules and why they exist

Division B and Division C share the construction section word for word. The table paraphrases Section 3 of the 2027 rules.

The safety parameters in 3.f are different from the rest. A vehicle that breaks any of them is not allowed to run until it is fixed, and if you cannot fix it during your competition time you receive participation points only. Other construction problems are scored as construction violations, each of which multiplies a run's scores by 0.8.

Speed control without a chip

Because integrated circuits are banned outside a commercial motor's own built-in electronics, a separate electronic speed controller is not allowed. That leaves a few legal ways to set thrust:

• Battery count. Changing the number of cells changes the voltage in steps (about 1.2 V per NiMH cell, 1.5 V per alkaline cell). Steps are coarse but repeatable.
• Series resistance. A rheostat or potentiometer in series with a motor lowers its current and speed. It wastes power as heat, and a small potentiometer can overheat at motor currents, so check its power rating.
• Propeller choice and pitch. A smaller or lower-pitch propeller gives less thrust at the same voltage.
• Thrust deflection. A vane that diverts part of the thrust air sideways or downward reduces forward thrust without touching the circuit.

Batteries, relays and timing mechanisms

With lithium banned, most teams use NiMH or alkaline cells. NiMH holds its voltage more steadily through a discharge, which helps repeatability, while alkaline voltage sags noticeably as the cell drains. Whatever you use, the label voltage decides the 12.0 V limit, not a meter reading, and the labels must be the manufacturer's originals.

Relays and mechanical switches open up simple automation. A relay lets one switch control a higher-current motor circuit. A mechanical switch tripped by another part of the vehicle, for example by a lever driven by a wind-up timer, could cut the thrust motor after a set time. Be careful with any shut-off scheme: a craft that stops moving for 3 seconds before the finish line makes the run incomplete, and the incomplete run loses its whole Time Score.

The run: from launch to finish

Knowing the sequence lets you plan the 8 minutes. The Event Supervisor checks the vehicle against the specifications before your competition time. If anything is out of spec, you are told as soon as possible and may fix it during your time. Then you receive the Target Time, and the 8-minute clock starts. Practice runs are not allowed.

• Prepare the track. You may dry-sweep the track and measure it and its surface. You may not wet it, put anything on it, or ask for it to be moved or re-leveled.
• Load and place. Put the vehicle, with its nickel load already on it, anywhere fully behind the start line. The Event Supervisor holds a wood block against the front of the craft. Switch on the motor or motors, then take your hands off the vehicle.
• Launch. Count down "3, 2, 1, launch". The Event Supervisor removes the block. Timing starts when the dowel crosses the start line and stops when it crosses the finish line.
• Hands off. From the moment your hands leave the vehicle at the start, you may not touch it or interfere with it in any way until the Event Supervisor says so. A touch or interference turns the run into a complete run with Distance, Time and Mass Scores of zero.
• Incomplete runs. A run is incomplete if the craft stops moving for 3 seconds before the finish line, or if the dowel has not crossed the finish within twice the Target Time. You cannot declare a run incomplete yourself. The Event Supervisor blocks the craft where the run was declared incomplete, you switch it off without moving it, and the distance from the finish line to the dowel is measured. Your clock pauses during the measurement.
• Not a run. If the dowel does not reach the start line within 3 seconds of launch, the attempt does not count as a run at all.
• Late starts. A run that begins before the 8 minutes are up is allowed to finish and is scored.
• Run limit. Competition time ends after two complete runs or two incomplete runs, or at 8 minutes, whichever comes first.
• Afterwards. The Event Supervisor reviews your scoresheet with you. To appeal, you must tell the Event Supervisor and leave the vehicle in impound.

Using the ramp is optional and may be decided run by run. The Event Supervisor places it in the same position for every team, and setting it up comes out of your 8 minutes. A ramp run has its Distance Score halved and its Mass Score set to zero, so the ramp is now a last resort for a craft that cannot start on the flat.

The track

The Event Supervisor supplies the track, so you build and calibrate against its stated dimensions. Build a faithful copy for practice.

The beams sit about 17 cm above the track and the top of your dowel must be at least 20 cm up, so the dowel always reaches beam height. What you must check is that nothing else on the craft, such as a propeller guard or a tall payload cover, reaches the beam before the dowel does. The tournament may run several tracks at once to keep teams moving.

Scoring

Scoring is the part teams most often misread, so this section works each formula with numbers. The formulas below are written in the manual's terms; check the scoring spreadsheet at soinc.org as well.

Notice what the Mass Score formula implies. A full roll of 40 nickels earns 4 points, which is 0.1 point per nickel, exactly the value of a loose nickel. So every nickel is worth 0.1 point however it is packaged, which is 0.02 point per gram, or 20 points per kilogram. Rolls are not worth more; they are simply far easier to hold securely than twenty loose coins.

The Time Score is symmetric: finishing 0.5 s early costs the same as 0.5 s late. Its sensitivity depends on the Target Time. At a 6.0 s target each 0.1 s of error costs $40 \times 0.1 / 6.0 = 0.67$ point. At 18.0 s it costs 0.22 point, and at 26.0 s only 0.15 point. Short targets are harsh and long targets forgiving, but long targets carry a greater risk of a stall.

Ties are broken in this order: the best Time Score from any run; then the fewest construction violations in the team's run that had the most of them (for a team with no runs, the count from the construction check); then the best Mass Score; then the second-best Run Score. The first tiebreaker means a precise Time Score matters even in a run that is not your best overall, and the third rewards carrying the load in every run.

Design principles for a repeatable loaded run

The design goal is a predictable average speed with the load you choose, using the power you can legally carry. Four principles drive most good designs.

Decide your load first. The full supply is worth 20 points, half a Distance Score. Because one dropped coin costs a 0.8 multiplier, the safest high-value loads are the ones you can hold mechanically. Four full rolls (16 points) plus the half roll (2 more) in fitted cradles is 18 points of low-risk mass. The last 2 points need twenty loose coins held in a closed tray or stacked slots. Design the vehicle around the load you will actually carry, then size the fan for it.

Keep the structure light and the cushion efficient. Lift pressure scales with total mass, and lift power with pressure to the 3/2 power. Every gram of frame you save is a gram of nickels you can carry for the same fan, or a larger lift margin for the same load. Use the largest cushion the 40 cm cube allows and place the load over its center.

Make thrust controllable, not large. The score rewards hitting a target average speed, so a stronger motor is not better. Excess thrust reaches speed early and makes the run harder to repeat. Pick a propeller, motor and battery combination whose output you can set repeatably with the legal controls (battery count, series resistance, propeller choice, thrust deflection), and calibrate it for each load.

Design the dowel and the payload cover together. The dowel must be within 3.0 cm of the front edge and lead the craft through the timing beam, and the payload holder must not stick up in front of it. A dowel that wobbles changes the timing, so make it a structural part of the frame. Make sure the propeller shields, which must stop a 3/8 inch dowel from reaching any blade, do not choke the airflow.

Testing logs and calibration

Practice runs are not allowed during competition time, and you may get as few as two runs, so all calibration happens before the tournament. Keep one row per practice run and compute the Run Score as you go. The table below is a sample log for a 10.0 s Target Time.

In run 1 the distance factor is $(185 - 37)/185 = 0.800$, applied to both the Distance and Mass Scores. Runs 3 and 4 show the loose-coin gamble: the best run of the day and a 20-point loss with the same configuration. A log like this tells you whether your holder is reliable enough to risk the last 2 points.

Record four things on every practice run: the load, the start position behind the line, the run time, and where any incomplete run stopped. Measure the lift margin at the start of each session. Repeatability is the real skill. If your complete runs with a given load are 9.2, 10.8 and 9.7 s, the average is close, but the spread is far too large to hit a half-second target at States.

Your impounded notes should include a calibration table for every Target Time your level can produce, for each load you might carry: the target average speed (1.85 m divided by the Target Time), the thrust setting, and the start position that gave that speed on your test track.

Strategy

Split the work by role. Both partners must be able to answer questions about the vehicle. A practical split is one student who loads, places and launches the craft and talks to the Event Supervisor, and one who keeps the run log, computes the score and checks the payload after each run. Swap roles in practice.

Plan for two runs. Competition time ends after two complete runs or two incomplete runs, so assume you will get two. Make the first run your safest scoring run: your best-calibrated load at the Target Time. Use the second either to repeat it or to add the riskier loose nickels if the first went cleanly.

Use the start position deliberately. A running start from 25 to 30 cm behind the line (for the loaded craft in the example above) removes launch lag from the timed section. Measure your own 3-second limit for each load and stay well inside it.

Check the track before your first run. Dry-sweep the lane, measure its width, and look for slopes or seams. You cannot ask for re-leveling, so if the surface slopes, adjust your thrust setting rather than the track.

Protect the impound and the safety check. A missed impound multiplies every run by 0.7, and a safety failure means participation points only. Before check-in, run a checklist from Section 3: the 40 cm cube with the load aboard, the dowel position and height, shielding on every propeller and impeller, original labels and no more than 12.0 V by label, no lithium or lead, no lasers, no integrated circuits outside a commercial motor, and an accessible manual switch on every motor. Impound your spare parts with the vehicle, and the batteries separately.

Know the appeal rule. To appeal you must tell the Event Supervisor and leave the vehicle in impound, so decide what you would dispute and how you would show it.

For the experimental discipline behind the logs, see the Experimental Design guide, which covers controls, repeated trials and variation. The Electric Vehicle guide covers another build event where the whole game is calibrating a run to a target, with the same logging and repeatability habits.

Units, vectors and measurement

Hovercraft tests mix units on purpose: a mass in grams, a track in centimeters, a speed in km/h and a force in pounds can all appear in one question. Convert everything to SI before you substitute, and the formulas in the rest of this guide will give answers in SI units automatically.

SI units for mechanics quantities

The SI has seven base units. Mechanics needs only three: the meter, the kilogram and the second. The other four are the ampere (current), kelvin (temperature), mole (amount of substance) and candela (luminous intensity). Every other mechanics unit is a derived unit built from m, kg and s. Write the derived unit out in base units when you want to check that an equation is dimensionally consistent: both sides must reduce to the same combination.

Notice that impulse and momentum have the same base units, so N·s and kg·m/s are both correct units for either. Torque and work also share base units, but by convention torque is written N·m and work J, because they are different kinds of quantity.

Metric prefixes

The kilogram is the odd base unit: it already carries a prefix. A mass given in grams must be divided by 1000 before you use it in $F = ma$ or $KE = \tfrac12 mv^2$. A 973 g block is 0.973 kg.

Converting units with the factor-label method

Multiply by fractions equal to 1, arranged so that the unwanted unit cancels. Squared and cubed units need the conversion factor squared or cubed: 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.

Significant figures and scientific notation

Scientific notation writes a number as $a \times 10^n$ with $1 \le a < 10$: 0.000 452 m = 4.52 × 10⁻⁴ m, and 1,109,000,000 J = 1.109 × 10⁹ J.

Significant figures show how precise a measurement is.

• All nonzero digits count: 4.52 has three.
• Zeros between nonzero digits count: 1002 has four.
• Leading zeros never count: 0.0045 has two.
• Trailing zeros count only if there is a decimal point: 2.50 has three, 2500 is ambiguous (write 2.50 × 10³ to show three), and "1000." with a decimal point has four.
• Multiplying or dividing: round the answer to the fewest significant figures among the inputs. 3.2 m ÷ 1.457 s = 2.2 m/s.
• Adding or subtracting: round to the fewest decimal places. 12.11 m + 3.4 m = 15.5 m.
• Exact counts and defined constants (2 in $\tfrac12 mv^2$, 1000 m in a km) do not limit precision.

Keep extra digits during a multi-step calculation and round only the final answer. Many tests accept an answer within a small tolerance (often a few percent) of the key, but a key written as "1.109 × 10⁹ J" is signalling that sig figs or notation may be graded.

Scalars and vectors

A vector is a quantity with both a magnitude (size) and a direction; it is drawn as an arrow whose length is the magnitude. A scalar has magnitude only. Scalars add like ordinary numbers; vectors add by geometry, so 3 N + 4 N can be anything from 1 N to 7 N depending on the angle between them.

Adding vectors

Tip to tail. Draw the first vector, start the second at the tip of the first, and the resultant (the vector sum) runs from the tail of the first to the tip of the last. The order does not matter. The parallelogram method gives the same answer.

Special cases.

• Same direction: add the magnitudes. 30 N east + 20 N east = 50 N east.
• Opposite directions: subtract, and the resultant points the way of the larger one. 50 N west + 25 N east = 25 N west.
• Perpendicular: use Pythagoras for the magnitude and the inverse tangent for the angle.

Components. Any vector of magnitude $A$ at angle $\theta$ above the +x axis splits into perpendicular parts. Add all the x-components, add all the y-components, then rebuild the resultant.

Relative velocity: boats, planes and currents

A velocity is always measured relative to something. A boat's velocity relative to the ground is its velocity relative to the water plus the water's velocity relative to the ground: $\vec v_{\text{boat/ground}} = \vec v_{\text{boat/water}} + \vec v_{\text{water/ground}}$. The same rule handles a plane in wind, a person walking on a train, and two cars on a highway.

• A tailwind adds to airspeed; a headwind subtracts. A plane with airspeed 220 km/h has ground speed 260 km/h in a 40 km/h tailwind and 180 km/h in a 40 km/h headwind.
• A crosswind adds at right angles: $\sqrt{220^2 + 40^2} = 224\ \text{km/h}$, drifting $\tan^{-1}(40/220) = 10.3°$ off course.
• Two objects moving in the same direction close at the difference of their speeds; moving toward each other, at the sum.

Kinematics

Kinematics describes motion (where, how fast, how quickly speeding up) without asking what causes it. Causes are forces, covered under Newton's laws below. Kinematics questions make up the largest calculation block in the Hovercraft bank, and nearly every one reduces to choosing the right equation.

Distance, displacement, speed and velocity

Distance is the total length of the path traveled, a scalar that never decreases. Displacement is the straight-line change in position from start to finish, a vector. Speed is how fast something moves (a scalar); velocity is speed with a direction (a vector).

Instantaneous velocity is the velocity at one moment: what a speedometer (plus a compass) reads. It equals the slope of the position–time graph at that instant. For motion at constant velocity, the instantaneous velocity at any moment equals the average velocity, so a craft that covers 60 miles at constant velocity between 2 p.m. and 4 p.m. is going 30 mph at 3 p.m.

The simplest equation, $d = vt$ at constant speed, also handles echo and signal problems. Sound or light makes a round trip, so the distance is doubled: a shout reflected from a cliff 170 m away returns after $2 \times 170 / 343 = 0.99\ \text{s}$ (343 m/s is the speed of sound in air at about 20 °C).

The average-of-speeds trap

The average speed for a multi-leg trip is total distance over total time. It is not the average of the leg speeds unless the legs take equal times.

Acceleration and sign conventions

Acceleration is the rate of change of velocity: $a = \Delta v / \Delta t$, in m/s² (meters per second, per second). Because velocity is a vector, an object accelerates if it speeds up, slows down or changes direction.

Choose a positive direction at the start of every problem and keep it. Then:

So "negative acceleration" does not always mean "slowing down", and "deceleration" means only that the acceleration points opposite the velocity. A car moving east and slowing has its acceleration pointing west. The velocity can be negative while the acceleration is positive, and the velocity can be zero while the acceleration is not.

The four constant-acceleration equations

These equations hold only when the acceleration is constant during the interval. Each one leaves out one of the five variables ($\Delta x$, $v_0$, $v$, $a$, $t$). List the three you know and the one you want, and pick the equation that skips the fifth.

Two shortcut patterns appear often:

• Halfway through a braking distance, the speed is $v_0/\sqrt2 \approx 0.71 v_0$, not half. Example: a boat at 24 m/s that stops in 90 m is still doing $24/\sqrt2 = 17.0\ \text{m/s}$ at the 45 m mark, because $v^2$ falls linearly with distance.
• From rest with constant acceleration, distances covered in successive equal time intervals are in the ratio 1 : 3 : 5 : 7 …, and total distance grows as $t^2$. An object that covers 1 m in the first 2 s has covered 4 m after 4 s and 9 m after 6 s.

Reaction time and stopping distance

A driver's total stopping distance has two parts. During the reaction time the vehicle keeps its original speed; then it brakes with roughly constant deceleration.

Meeting and chase problems

Write a position equation for each object using the same origin and clock, then set the positions equal.

Free fall under gravity

An object in free fall moves under gravity alone. Near Earth's surface, with air resistance neglected, every object in free fall has the same downward acceleration $g = 9.81\ \text{m/s}^2$ (tests may use 9.8 or 10). A feather and a coin dropped in an evacuated tube land together, and a 1 kg binder and a 2 kg broken hovercraft dropped in a vacuum fall side by side. Heavier objects feel a larger gravitational force, but they have proportionally more inertia, so $a = F/m = mg/m = g$ is the same for all.

In air, drag changes this: a feather and a sheet of paper fall far slower than a bowling ball, so "two objects dropped from the same height in air always land together" is false. Drag and terminal velocity are covered in the Friction and drag chapter.

Halfway down. An object dropped from height $H$ has, at height $H/2$, a speed of $v_{\text{final}}/\sqrt2$ and half its final kinetic energy. Halfway through the time, it has fallen only one quarter of the height.

Thrown straight up

For an object thrown upward, gravity is still the only force. Take up as positive; then $a = -g$ for the entire flight, on the way up, at the top, and on the way down.

• At the top, the velocity is zero but the acceleration is still 9.81 m/s² downward. If the acceleration were zero there, the ball would hover.
• The motion is symmetric (no air resistance): the time up equals the time down, and the ball passes every height with the same speed going up and coming down. It returns to the launch point at its launch speed.
• Maximum height depends on launch speed squared: double the speed, four times the height.

Motion graphs

Tests ask you to read a graph or to match a motion to its graph. Three rules cover almost everything:

• The slope of a position–time (x–t) graph is the velocity. A curved x–t graph means changing velocity.
• The slope of a velocity–time (v–t) graph is the acceleration; the area between the v–t line and the time axis is the displacement (area below the axis counts as negative).
• The area under an acceleration–time (a–t) graph is the change in velocity.

Projectile motion in two dimensions

A projectile is any object moving under gravity alone after launch (air resistance neglected). Its path is a parabola. The key idea is that horizontal and vertical motions are independent:

• Horizontally there is no force, so the horizontal velocity $v_x$ is constant and the horizontal acceleration is zero.
• Vertically the object is in free fall with acceleration $g$ downward.
• Time is the only quantity the two directions share. Solve the vertical motion for the time, then use it horizontally.

A bullet fired horizontally and a bullet dropped from the same height at the same moment hit level ground at the same time. A projectile has one force on it in flight (gravity), so its acceleration is constant even though its velocity changes direction.

Horizontal launch from a height

Angled launch on level ground

Split the launch velocity into components: $v_{0x} = v_0\cos\theta$ and $v_{0y} = v_0\sin\theta$, with $\theta$ above the horizontal.

Facts tests test directly:

• 45° gives the maximum range on level ground, because $\sin 2\theta$ peaks at 90°.
• Complementary angles give equal ranges. 30° and 60° (or 20° and 70°) land in the same place; the steeper one goes higher and stays up longer.
• 90° gives the maximum height for a given speed (and zero range).
• Height, range and time all scale with launch speed: $H$ and $R$ go as $v_0^2$, $T$ as $v_0$. Four times the launch speed means sixteen times the maximum height.
• The time in the air is set entirely by the vertical motion.

Launching from a cliff at an angle

When the landing point is lower than the launch point, the level-ground formulas fail. Write the vertical equation and solve the quadratic for time.

What stays constant during flight

At the top of the path the velocity (horizontal) and the acceleration (vertical) are perpendicular; that is the only point where this happens. Asking "where is the acceleration greatest?" is a trap: it is the same everywhere.

Dropping from a moving vehicle and the monkey-and-hunter

A package released from a plane flying level keeps the plane's horizontal velocity. With no air resistance it stays directly below the plane as it falls: the pilot sees it drop straight down, while a person on the ground sees a parabola. A crate falling from a plane above a car therefore lands where the plane is at impact, not on the car below the release point. The same idea applies to a ball tossed straight up inside a train moving at constant velocity: the passenger sees it go straight up and down, and an observer on the platform sees a parabola.

Monkey and hunter. A hunter aims directly at a monkey hanging from a branch; the monkey lets go the instant the gun fires. Both the bullet and the monkey fall the same $\tfrac12 gt^2$ below their gravity-free paths, so the bullet hits the monkey (if it reaches the monkey's horizontal position before hitting the ground). The lesson: aim straight at a target that starts falling at launch.

Hitting a target moving overhead. A projectile fired at the instant a target flying horizontally at constant speed $v_t$ passes directly overhead stays directly under the target only if its horizontal velocity equals the target's. So aim ahead, tilted from vertical by $\theta$ where $v\sin\theta = v_t$. Gravity changes only the vertical motion, so the projectile hits as long as its vertical launch speed can carry it up to the target's height $h$, that is, $v\cos\theta \ge \sqrt{2gh}$. It meets the target on the way up.

Newton's laws of motion

Dynamics connects motion to its cause, which is force. A force is a push or a pull, measured in newtons (N); one newton accelerates 1 kg at 1 m/s². Isaac Newton published the three laws in 1687 (Principia), and they still describe everything a hovercraft does on the track.

First law: inertia

Inertia is the tendency of an object to resist any change in its motion. Mass is the measure of inertia: a school bus at rest has far more inertia than a fast compact car, because inertia depends on mass, not on speed.

The first law says that if the net force (the vector sum of all forces) on an object is zero, its velocity does not change: it stays at rest, or it keeps moving in a straight line at constant speed. An object with zero net force is in equilibrium: static equilibrium if at rest, dynamic equilibrium if moving at constant velocity.

Everyday first-law examples:

• A car stops suddenly and loose objects in the back seat fly forward. Nothing pushes them forward; they keep moving while the car stops. Seat belts supply the backward force that stops your body with the car.
• Standing in a bus, you lurch forward when the bus brakes and backward when it starts.
• A ball in a wagon rolls toward the back when the wagon is jerked forward. Relative to the ground, the ball stays nearly where it was; the wagon moves under it.
• A tablecloth yanked quickly leaves the dishes behind.
• A rocket in deep space with its engine off coasts at constant velocity forever.
• A skydiver at terminal velocity has drag equal to weight, so net force is zero and the speed is constant.
• A hovercraft keeps gliding after its thrust fan stops, because the cushion leaves very little friction to slow it down.

Second law: net force and acceleration

Proportional reasoning appears in many questions. Doubling the net force doubles $a$; doubling the mass halves it. Doubling both the forces and the mass leaves the acceleration unchanged. A net force $2F$ on a mass $4m$ gives $a/2$; a net force $4F$ on $2m$ gives $2a$.

Mass and weight

Mass is the amount of matter in an object and the measure of its inertia; it is a scalar in kilograms and is the same on Earth, on the Moon and in orbit. Weight is the gravitational force on the object, a vector in newtons pointing toward the planet's center. Saying "the amount of matter in an object is its weight" is false; that is mass.

The astronaut's mass is 70 kg in every row. A bowling ball with mass 10 kg on Earth has mass 10 kg on any planet. Bathroom scales that read in kilograms actually measure force and divide by 9.81, so they read wrong on the Moon.

Weightlessness in orbit is free fall, not zero gravity. At the altitude of the International Space Station, gravity is still about 90% as strong as at the surface. Astronauts feel weightless because they and the station fall around Earth together with the same acceleration, so nothing pushes on them. A satellite in a circular orbit at constant speed therefore has a nonzero net force on it: gravity, pointing toward Earth's center, supplies its centripetal acceleration.

Free-body diagrams and the normal force

A free-body diagram (FBD) shows one object as a dot or box with an arrow for every force on that object, each drawn from the object and labeled. Leave out forces the object exerts on other things, and never draw "ma" or "the force of motion" as a force. Then choose axes (along and perpendicular to the motion, or along an incline) and apply $\Sigma F = ma$ along each.

The normal force $N$ is the push of a surface on an object, perpendicular to the surface. It is whatever size is needed to stop the object from sinking into the surface, so it is not always equal to mg:

• On a level floor with only vertical forces and no vertical acceleration: $N = mg$.
• Pushing down on a box at an angle: $N = mg + F\sin\theta$ (bigger).
• Pulling up on a box at an angle: $N = mg - F\sin\theta$ (smaller).
• On an incline: $N = mg\cos\theta$.
• In an accelerating elevator: $N = m(g + a)$.

A person sitting still on a chair has two forces, weight down and the chair's normal force up, and the resultant force on them is zero. Lifting one foot while standing on a scale changes nothing: the scale still supports the whole weight.

Apparent weight in elevators and rockets

A scale measures the normal force it pushes up with, which tests call your apparent weight. Take up as positive. Two forces act on you: $N$ up and $mg$ down, so $N - mg = ma$.

Third law: action and reaction

Forces always come in pairs. If object A exerts a force on object B, then B exerts a force on A that is equal in size and opposite in direction. The two forces of a third-law pair:

• act on different objects;
• are the same type of force (both contact, both gravitational, and so on);
• appear and disappear together;
• are equal in size even when the masses are very different.

When a hammer hits a feather, a truck hits a mosquito, or a heavy football player collides with a light one, each pair of forces is equal. The effects differ because the same force produces a large acceleration in a small mass and a small acceleration in a large mass ($a = F/m$). A meteor pulls on Earth exactly as hard as Earth pulls on the meteor; Earth's acceleration is just immeasurably small.

Why don't the pair forces cancel? Forces cancel only when they act on the same object. The action force acts on one body and the reaction on another, so each body's motion is decided by the forces on it alone.

Horse and cart. A horse argues it cannot pull the cart, because the cart pulls back with an equal force. The flaw: the cart's backward pull acts on the horse, while the horse's forward pull acts on the cart. The cart accelerates if the horse's pull exceeds the friction on the cart. The horse accelerates because the ground pushes it forward (friction on its hooves, the reaction to its hooves pushing backward on the ground) harder than the cart pulls it back.

Book on a table. Two forces act on the book: Earth's gravity down ($mg$) and the table's normal force up ($N$). They are equal and opposite, but they are not a third-law pair, because both act on the book; they are equal because of the first law (no acceleration). The true pairs are: Earth pulls the book down ↔ the book pulls Earth up; the table pushes the book up ↔ the book pushes the table down.

A rocket does not need air to push against. It pushes its own exhaust backward, and the exhaust pushes it forward, which is why rockets work in space.

Contact and non-contact forces

Contact forces act only when surfaces touch: normal force, friction, tension, air resistance (drag), the buoyant force, the push of a cushion of air, and applied pushes and pulls. Non-contact forces act at a distance through a field: gravity, the electric force and the magnetic force. "Which is not a contact force?" almost always has gravity as the answer.

At the atomic level every contact force is electromagnetic: atoms in the two surfaces repel each other when pushed close. All forces in nature reduce to four fundamental forces:

Gravity is the weakest force between two particles but dominates at large scales because masses do not cancel the way positive and negative charges do.

Newton's laws conceptual traps

Which law explains it?

Newton's laws on a hovercraft

Lift and thrust. A hovercraft uses two forces from air: lift from cushion pressure under the deck, which balances the weight, and thrust from air pushed backward. Both are third-law reactions to the fans pushing air. With lift equal to weight, the vertical net force is zero and the craft floats; the horizontal motion is then set by thrust minus drag and residual friction.

Why it keeps gliding. Because the cushion removes almost all contact friction, very little force opposes the motion. When the thrust stops, the first law says the craft keeps going, slowing only gradually under air drag and skirt friction. This is also why stopping and steering are hard: a car turns because the road pushes sideways on its tires, but a hovercraft has almost no sideways grip, so it slides in the direction it was already going (more on turning under the circular-motion section below).

Why it veers or spins.

• Off-center thrust: if the thrust line does not pass through the center of mass, the thrust also produces a turning effect (torque) and the craft yaws.
• Weight imbalance: if the load sits to one side, that side presses harder, its skirt gap closes and it drags on the track, while the light side lifts and leaks air. The extra friction on one side turns the craft toward the heavy side. Fix it by moving the load or ballast toward the center.
• Propeller torque reaction: a spinning propeller or fan pushes the air around one way, and by the third law the air twists the craft the other way. Counter-rotating fans or guide vanes cancel this.

Pitching when it starts. When thrust first accelerates the craft, the thrust acts along the propeller axis while the craft's inertia effectively acts at its center of mass. If the two lines are at different heights, the pair makes a turning effect that tips the craft. A thrust line below the center of mass raises the nose and lowers the back, the way a car squats when its tyres push it forward at road level. Skirt friction pulling backward at the bottom has the opposite effect and tends to tip the nose down. Once the craft reaches constant velocity, thrust balances drag and a well-balanced craft rides level.

Forces in two dimensions

Inclined planes

On a slope, choose axes along the incline and perpendicular to it. Then the normal force and any friction lie along axes, and only the weight needs splitting. The angle between the weight vector and the perpendicular to the slope equals the incline angle θ.

If a test gives the incline as a right triangle with a 4 m base and 3 m height, the slope is 5 m long and $\sin\theta = 3/5 = 0.6$, $\cos\theta = 4/5 = 0.8$; you do not need the angle itself (36.9°).

When does a resting object start to slide? Raise one end of a plank slowly. The slope force $mg\sin\theta$ grows and the maximum static friction $\mu_s mg\cos\theta$ shrinks. Sliding begins when they are equal, at $\tan\theta = \mu_s$, whatever the mass: $\mu_s = 0.45$ gives $\theta = \tan^{-1}0.45 = 24.2°$. The formula and a second example are in The angle of repose.

Tension, strings and pulleys

Tension is the pulling force a string, rope or cable exerts along its length. Problems assume an ideal string: massless and unstretchable. Then the tension is the same at every point along it, and everything tied to it moves with the same speed and acceleration magnitude. An ideal pulley is massless and frictionless; it changes the direction of the tension without changing its size.

In a tug-of-war where each team pulls with 100 N and nobody moves, the tension in the rope is 100 N, not 200 N: each end of the rope is pulled with 100 N, just as a rope tied to a wall and pulled with 100 N has 100 N of tension.

Atwood machine. Two masses hang on either side of a pulley. The heavier side falls and the lighter rises with the same acceleration magnitude. The driving force is the difference in weights, and it must accelerate both masses.

Block on a table with a hanging mass. A block $m_1$ on a horizontal table is tied over a pulley at the edge to a hanging mass $m_2$. Only the hanging weight drives the system (minus any friction on the table block).

Strings at angles holding a weight. For an object hanging at rest from two strings, the vertical components of the tensions add to the weight, and the horizontal components cancel.

Forces applied at an angle

A pull at angle θ above the horizontal does two jobs: $F\cos\theta$ moves the object forward, and $F\sin\theta$ lifts on it, which reduces the normal force and therefore the friction. A push at angle θ below the horizontal does the opposite: it presses the object into the floor and increases friction.

Uniform circular motion

Uniform circular motion is motion in a circle at constant speed. The speed is constant but the velocity is not, because its direction keeps changing. So the object is accelerating, and the acceleration points toward the center of the circle: the centripetal ("center-seeking") acceleration. The velocity at every instant is tangent to the circle, at right angles to the acceleration.

The centripetal force is not a new kind of force. It is the name for whichever real force, or combination, points toward the center: tension in a whirled string, friction for a car on a flat curve, gravity for a satellite, the normal force for a rider pressed against a spinning wall. Never add "centripetal force" as an extra arrow on a free-body diagram.

Centrifugal force is fictitious. When a car turns sharply, you feel pushed toward the outside of the turn. No outward force acts on you: your body tends to keep going straight (first law) while the car turns inward beneath you, until the door or seat pushes you inward. "Centrifugal force" only appears if you describe the motion from inside the rotating car.

When the string breaks, the centripetal force disappears and the object moves off in a straight line along the tangent at the point of release, with the speed it had at that instant. It does not fly outward along the radius. After release it becomes an ordinary projectile.

Tension in a horizontal whirl does no work, because it is always perpendicular to the velocity: it changes the direction of motion, not the speed (see Work, energy and power below).

Flat and banked curves

On a flat road, static friction between the tires and road provides the centripetal force. The fastest safe speed is when the needed force equals the maximum static friction, $mv^2/r = \mu_s mg$.

The same angle appears when a cyclist or motorcyclist leans into a turn. Friction on the tires supplies the centripetal force $mv^2/r$ and the road pushes up with $mg$. For the rider not to topple, the total road force must point through the center of mass, so the lean angle from vertical satisfies $\tan\theta = \dfrac{v^2}{rg}$, the same formula as the ideal banking angle, and it needs a friction coefficient of at least $\tan\theta$.

Vertical circles, hills and loops

In a vertical circle the centripetal force is the net inward force, combining weight with tension or normal force.

• Bottom of a vertical circle: $T - mg = mv^2/r$, so $T = mg + mv^2/r$. The string is most likely to break here.
• Top of a vertical circle: $T + mg = mv^2/r$. The minimum speed to stay on the circle is when $T = 0$: $v_{\min} = \sqrt{gr}$. The same result holds for a loop-the-loop track.
• Crest of a hill: $mg - N = mv^2/r$, so $N = mg - mv^2/r$, less than the weight; the car feels light. At $v = \sqrt{gr}$ the normal force reaches zero and the car leaves the road.
• Bottom of a dip: $N = mg + mv^2/r$, more than the weight.

Conical pendulum and the rotor ride

A conical pendulum is a mass on a string of length $L$ swinging in a horizontal circle, with the string at angle θ from the vertical. The vertical part of the tension holds up the weight ($T\cos\theta = mg$) and the horizontal part supplies the centripetal force ($T\sin\theta = mv^2/r$), with $r = L\sin\theta$. Dividing: $\tan\theta = v^2/(rg)$.

In a rotor ride, riders stand against the wall of a spinning cylinder and the floor drops away. The wall's normal force points inward and supplies the centripetal force, $N = mv^2/r$; static friction on the wall holds up the weight, so $\mu_s N \ge mg$. The minimum coefficient is $\mu_s = gr/v^2$, independent of the rider's mass.

Work, energy and power

Energy is the capacity to do work; it is a scalar, measured in joules, and it is never created or destroyed, only transferred or transformed (the law of conservation of energy). Energy and momentum are the two great bookkeeping tools of mechanics; momentum and impulse are covered in the Momentum, impulse and collisions chapter.

Work done by a force

Work is the energy transferred to or from an object by a force acting through a displacement. Only the part of the force along the displacement does work.

Zero-work cases. Holding a heavy box still (no displacement), pushing on a wall that does not move, carrying a box horizontally at constant velocity (your upward force is perpendicular to the motion), the normal force on a block sliding along a floor, and the centripetal force in uniform circular motion (always perpendicular to the velocity, so the speed never changes). Your muscles tire while holding a box, but that is chemical energy turned to heat inside you, not work on the box.

Negative work removes energy. Kinetic friction does negative work on a sliding object, turning its kinetic energy into thermal energy. Gravity does negative work on a ball rising and positive work on it falling.

Work by gravity depends only on the height change, not the path. Two cyclists climbing the same mountain, one straight up a steep road and one on a long winding road, do the same work against gravity ($mgh$, for equal masses). The steep route needs more force over a shorter distance; if it is climbed in less time, it needs more power.

Machines do not save work. A ramp, lever or pulley system lets you use a smaller force over a longer distance. In the ideal case the work is the same as lifting directly; with friction, a machine needs more work, never less.

Kinetic energy and how it scales

Kinetic energy is the energy of motion.

Because speed is squared, speed has the greater influence: doubling the speed quadruples the KE, while doubling the mass only doubles it. Tests love ratio problems. Multiply the mass factor by the square of the speed factor:

Two objects with equal KE have equal momenta only if their masses are equal (and they move the same way). With equal KE, the lighter object is the faster one: a nail with half a hammer's KE but a hundredth of its mass still moves faster.

The work-energy theorem

Gravitational potential energy

Potential energy is stored energy due to position or configuration. Gravitational potential energy near Earth's surface is the weight times the height above a chosen reference level.

Elastic potential energy and springs

An ideal spring obeys Hooke's law: the force it exerts is proportional to how far it is stretched or compressed from its natural length, and points back toward that length (a restoring force).

A falling bungee jumper is fastest not when the cord first goes taut but a little later, where the cord's upward force $kx$ has grown to equal the weight $mg$. Beyond that point the net force points up and the jumper slows.

Conservation of mechanical energy

Mechanical energy is $KE + PE$. When only conservative forces (gravity, ideal springs) do work, mechanical energy stays constant: what one form loses, the other gains.

Pendulum. At the highest points of the swing the bob is momentarily at rest: PE is maximum and KE is zero. At the lowest point PE is minimum and KE (and speed) is maximum. Halfway down in height, the energy is half PE and half KE.

Energy lost to friction

When friction or air resistance acts, mechanical energy is not conserved: some becomes thermal energy (and some sound). Total energy is still conserved.

Conservative and non-conservative forces

Power and horsepower

Power is the rate of doing work or transferring energy.

At constant power, the available force falls as speed rises: $F = P/v$. A motor with fixed power output that runs three times as fast can push with only one third of the force, so it produces one third of the acceleration on the same mass.

Energy from power. Energy = power × time. A 1.5 kW heater running 3.0 h uses $1.5 \times 3.0 = 4.5\ \text{kWh}$, which is $4.5 \times 3.6 \times 10^6 = 1.6 \times 10^7\ \text{J}$. The kilowatt-hour is an energy unit, not a power unit.

Efficiency of machines and motors

No real machine turns all its input energy into useful output; the rest becomes heat, sound or vibration.

On a hovercraft, the chain is battery (chemical energy) → electrical energy → motor (losses as heat) → fan or propeller (kinetic energy of air, with more losses) → thrust and cushion. Every stage is less than 100% efficient, which is why a measured run uses far more battery energy than the craft's small kinetic energy suggests.

Units of energy

Work and energy traps

Friction and drag

Friction and drag are both resistive forces: they act against the motion of an object relative to whatever it touches. Friction comes from two solid surfaces in contact. Drag comes from a fluid (a gas or a liquid) flowing past an object. Tests ask about both, often in the same hovercraft story, because a hovercraft's whole purpose is to trade the first for the second.

Static and kinetic friction

Real surfaces are rough on a microscopic scale. Where two surfaces touch, the high points press together and partly bond, so it takes a force to shear them apart. Friction is the contact force parallel to the surfaces that results.

• Static friction acts when the surfaces are not sliding relative to each other. It is a "responsive" force: it takes whatever value is needed to stop sliding, from zero up to a maximum. Push a heavy crate gently and it does not move, because static friction exactly matches your push. Push harder and static friction grows to match, until it reaches its limit $\mu_s N$.
• Kinetic friction acts once the surfaces slide. For ordinary speeds it is close to constant, $f_k = \mu_k N$, whatever the push.

Here $N$ is the normal force, the perpendicular push between the surfaces, and $\mu$ (mu) is the coefficient of friction, a dimensionless number that depends on the two materials. Because the bonds have time to form while the surfaces sit still, $\mu_s$ is usually larger than $\mu_k$, which is why it is harder to start a crate sliding than to keep it sliding.

Four facts about friction are tested again and again:

1. Direction. Friction always opposes the relative motion (kinetic) or the tendency of relative motion (static) between the two surfaces. That is not always opposite to the object's motion. When you walk, your foot pushes backward on the ground and static friction pushes you forward. A box on the floor of a truck that speeds up is carried forward by static friction.
2. Area. To a first approximation, friction does not depend on the area of contact. A brick slides with the same friction on its face or on its edge: a larger area spreads the same normal force more thinly.
3. Speed. Kinetic friction is roughly independent of sliding speed at everyday speeds.
4. Normal force, not weight. Friction is proportional to $N$, which equals $mg$ only on a level surface with no vertical push or pull. On the Moon, where $g$ is about one sixth of Earth's, the same box has one sixth the normal force and one sixth the friction.

The coefficients below are typical values. Values vary between sources and between samples, because surface finish, cleanliness and moisture matter a great deal, so treat them as guides, not constants.

Values from the OpenStax physics tables (see References); other sources differ by tens of percent.

Pushing or pulling at an angle

A pull at an angle above the horizontal does two things: its horizontal part drives the object forward, and its vertical part lifts a little, which reduces the normal force and so reduces friction. A push angled downward does the opposite, pressing the object into the floor and increasing friction. (Resolving a force into components is covered in "Forces in two dimensions".)

The angle of repose

Tilt a board with a block on it, slowly. The block starts to slide at the angle where the downhill component of gravity, $mg\sin\theta$, just equals the maximum static friction, $\mu_s mg\cos\theta$. The mass cancels:

Rolling versus sliding

A wheel that rolls without slipping has no sliding at the contact point, so the friction there is static, and it does no work to slow the wheel. What slows a rolling wheel is rolling resistance, mostly energy lost as the tire or the surface deforms. Rolling-resistance coefficients are small, roughly 0.001 to 0.002 for steel train wheels on rails and about 0.01 for car tires on pavement, far below sliding values. That is why wheels were invented, and why a skidding (sliding) car stops in a different distance than one braking just short of a skid: static friction (no skid) can exceed kinetic friction (skid), which is the idea behind anti-lock brakes.

Why hovercraft cut friction

A hovercraft does not slide on its hull. The cushion pressure carries the weight, so the normal force between hull and ground is almost zero, and so is contact friction. What remains is air drag, the drag of any skirt that brushes the surface, and, over water, the drag of the spray and the shallow depression the cushion pushes into the water. The flip side of almost no friction is almost no grip: a hovercraft cannot use friction to steer or brake, so it drifts sideways in a turn and stops slowly. "How real hovercraft work: history and design" covers how real craft deal with this.

Stopping distance with friction

When only kinetic friction slows an object on a level surface, the deceleration is $a = \mu_k g$ (the mass cancels). Combining with $v^2 = v_0^2 - 2ad$ gives the stopping distance.

Block on block

When one block sits on another, friction between them is what drags the top block along. The top block can only accelerate as fast as the maximum static friction allows: $a_{\max} = \mu_s g$. If you push the bottom block harder than that, the top block slips.

When the push is larger than that limit, the blocks slide relative to each other and kinetic friction acts at the interface. Treat each block separately. The friction on the top block points forward (it is dragged along); the same-sized friction on the bottom block points backward (Newton's third law). If the floor also has friction, the floor's normal force on the bottom block is the weight of both blocks, because the bottom block carries the top one.

Air resistance and the drag equation

When an object moves through a fluid, it has to push fluid out of its way and drag fluid along its surface. The fluid pushes back with a drag force that points opposite to the object's velocity relative to the fluid. For most everyday objects (a ball, a car, a person, a hovercraft) moving at ordinary speeds, drag is described by the drag equation.

Read the equation as a list of what drag depends on: the fluid's density, the speed squared, the shape, and the frontal area. It does not depend on mass. That leads to the scaling questions tests love:

• Double the speed → drag × 4. Triple → × 9.
• Double the radius of a sphere or circular disk → area × 4 → drag × 4.
• Move from air to water (about 830 times denser) → drag × 830 at the same speed and shape.
• The power needed to push through drag is $P = F_D v \propto v^3$, so doubling a vehicle's cruising speed needs about eight times the power.

The drag coefficient captures everything about shape. Typical values (from the OpenStax table; real values depend on the exact shape and the Reynolds number, defined below):

Skin friction, pressure drag and streamlining

Drag has two main parts.

• Skin friction drag comes from the fluid shearing along the surface, the viscous "rubbing" in the thin layer next to the surface (the boundary layer, below). It grows with the wetted surface area.
• Pressure drag (also called form drag) comes from the flow separating from the back of a blunt body, leaving a low-pressure, turbulent wake. High pressure on the front and low pressure behind push the object backward. For a blunt object it is by far the larger part.

Streamlining means shaping a body (rounded nose, long gently tapering tail, like a teardrop) so the flow stays attached and the wake stays small. It slashes pressure drag at the cost of a little more surface. A third kind, induced drag, is the drag that comes with producing lift on a wing: tilting the airflow downward to make lift also tilts the force backward, and wingtip vortices carry that energy away. It matters for aircraft and wing-in-ground-effect craft, not for a hovercraft on a track.

Terminal velocity and falling with drag

Drop an object from rest. At first it is slow, drag is negligible, and it accelerates at nearly $g$. As it speeds up, drag grows with $v^2$, so the net force $mg - F_D$ shrinks, and the acceleration shrinks with it. Eventually drag equals weight, the net force is zero, and the object falls at a constant speed, the terminal velocity $v_t$. By Newton's first law, nothing more changes after that.

The phases of a skydive follow from the same balance:

Several conceptual results come straight from the formula:

• Heavier falls faster in air, for the same shape and size. $v_t \propto \sqrt{m}$, so a steel ball outruns a same-size plastic ball in air. A heavier object must go faster before drag can balance its larger weight.
• In a vacuum, everything falls together. With no air, a feather and a coin, or a bowling ball and a sheet of paper, have the same acceleration $g$ and land together. In air, the feather's tiny mass and large area give it a very low terminal velocity.
• At terminal velocity the velocity is not zero. The acceleration is zero.
• Thrown upward with air resistance, an object takes longer to come down than to go up and lands slower than it was thrown, because drag acts against the motion on both legs.

The same balance governs a hovercraft at cruise: thrust plays the role of weight and drag plus friction grow until they match it, so the craft settles at a "terminal" cruise speed.

Viscosity: a fluid's internal friction

Viscosity is a fluid's internal friction: its resistance to flowing, or to one layer sliding over another. Honey is more viscous than water, and water more viscous than air. A fluid with zero viscosity is called inviscid; the "ideal fluid" of Bernoulli's equation is inviscid.

Picture a thin layer of fluid of thickness $d$ between two plates. Hold the bottom plate still and drag the top plate (area $A$) at speed $v$. The fluid touching each plate moves with that plate, and the layers in between shear past one another. The force needed is:

Temperature. The viscosity of a liquid falls as it warms (warm syrup pours easily), because the molecules escape each other's attraction more easily. The viscosity of a gas rises as it warms, because gas viscosity comes from molecules carrying momentum between layers, and hotter molecules move faster. Pressure. Very high pressure increases the viscosity of liquids (strongly for oils); for gases the effect is small over ordinary ranges.

Viscous flow in a pipe (Division C). Viscosity makes the fluid at a pipe wall stand still and the fluid in the center move fastest. Poiseuille's law gives the volume flow rate of smooth (laminar) flow through a pipe of radius $r$ and length $L$ under a pressure difference $\Delta P$: $Q = \dfrac{\pi r^4 \Delta P}{8\eta L}$. The fourth power is the headline: double the pipe's diameter and, at the same pressure difference, the flow rate rises 16 times. Equivalently, to keep the same flow through a narrower pipe takes a much larger pressure difference. In very narrow tubes viscosity dominates, which is why Bernoulli's equation (which ignores viscosity) works poorly there.

Newtonian and non-Newtonian fluids

A Newtonian fluid has a viscosity that stays the same however fast it is stirred or sheared: water, air, gasoline and most thin oils behave this way, and every viscosity formula in this chapter assumes it. A non-Newtonian fluid has a viscosity that changes with the shear rate (how hard or fast it is deformed).

The classic test question: you can run across a pool of cornstarch suspension but sink if you stand still, because the sudden impacts make it shear-thicken.

Stokes' law for small spheres

For a small sphere moving slowly through a viscous fluid (very low Reynolds number), drag is proportional to speed, not speed squared.

Reynolds number, laminar and turbulent flow

Fluid flow comes in two broad types.

• Laminar flow is smooth and orderly: the fluid moves in parallel layers (streamlines) that do not cross or mix. Each particle follows a steady path.
• Turbulent flow is irregular and chaotic, full of swirling eddies; layers mix and the velocity at a point fluctuates.

Which one you get depends on the competition between the fluid's inertia (its tendency to keep going, which breaks the flow into eddies) and its viscosity (which damps disturbances and keeps layers orderly). The Reynolds number is the ratio of the two.

For flow in a round pipe, flow is usually laminar below $Re \approx 2000$ to $2300$ and turbulent above about $3000$ to $4000$, with an unsteady transition range in between. Sources quote slightly different cut-offs, so read the numbers a test gives. When $Re \ll 1$, viscous forces dominate and inertia can be ignored: bacteria swimming, dust settling, pollen in air, slow flow through porous soil, and Stokes' law. When $Re$ is huge (aircraft, ships, wind on buildings), inertia dominates and flow is mostly turbulent except in thin layers near surfaces.

Boundary layer. Right at a solid surface the fluid does not slip; its speed is zero relative to the surface. Speed rises to the free-stream value across a thin boundary layer. Skin-friction drag happens in this layer. The boundary layer usually starts laminar near the front of a body and becomes turbulent farther back. A turbulent boundary layer clings to a curved surface longer before separating, which is why the dimples on a golf ball, by triggering turbulence in the boundary layer, shrink the wake and reduce total drag.

In a hovercraft, the same ideas explain why designers round the corners of a cushion and skirt: sharp corners and sudden changes in a duct cause separation and turbulence, which waste the fan's energy. A smooth, rounded plenum and duct keep losses down and spread the pressure evenly.

Momentum, impulse and collisions

Momentum as a vector quantity

Momentum is mass times velocity, $\vec p = m\vec v$. It is a vector: it has the direction of the velocity. Its SI unit is the kg·m/s, which is the same as a newton-second (N·s), since 1 N = 1 kg·m/s². Momentum is what Newton called the "quantity of motion": a slow truck and a fast bullet can have the same momentum.

Scaling questions are common: momentum is proportional to both mass and speed. Twice the mass at the same speed doubles $p$. Three times the mass at half the speed gives 1.5 times $p$. One third the mass at four times the speed gives 4/3 times $p$. Compare kinetic energy, $KE = \tfrac{1}{2}mv^2 = p^2/(2m)$: doubling the speed doubles $p$ but quadruples $KE$.

At the top of a vertical throw the velocity, and so the momentum, is momentarily zero. A ball moving in a circle at constant speed has constant-size momentum whose direction keeps changing, so its momentum is not constant.

Impulse and the impulse-momentum theorem

A force acting over a time changes momentum. The product of the average force and the time is the impulse.

The theorem is Newton's second law multiplied through by $\Delta t$, so anything that obeys $F = ma$ obeys it too.

Bouncing gives a larger impulse than stopping

If an object stops, its momentum changes by $mv$. If it bounces back, its momentum changes by $mv$ to stop plus $mv'$ to come back: $\Delta p = m(v + v')$. A bounce therefore needs a bigger impulse, and for the same contact time a bigger force. That is why a rubber ball knocks over a block that a clay ball of the same mass and speed does not, and why a turbine blade that turns water right around (a Pelton wheel) gets more push than a flat paddle.

Making the stopping time longer

For a fixed change in momentum (you are going to stop either way), $F_{\text{avg}} = \Delta p/\Delta t$, so stretching the stopping time lowers the force. This one idea explains:

• Airbags and seat belts, which spread the occupant's stop over a longer time (and a larger area).
• Crumple zones, which make the front of a car collapse in a controlled way so the passenger compartment stops more gradually. A car designed for safety should have as long a collision time as possible.
• Catching an egg or a water balloon by moving your hands back with it, landing with bent knees, rolling when you fall, padded gym mats and the cushioned barrier on a Hovercraft track.
• Follow-through in a golf or tennis swing, which keeps the club in contact longer and so delivers a larger impulse for the same force, sending the ball off faster.

A sudden jerk can also be used on purpose: the hanging-weight demonstration in First law: inertia works because a very short pull delivers too small an impulse to move the heavy mass, so the lower thread breaks before the extra tension reaches the upper one.

Thrust from a stream of fluid

A fan, propeller or rocket gets its push by changing the momentum of a stream of air or exhaust each second. The thrust equals the momentum carried away per second. This is the rocket-thrust formula from Third law: action and reaction, now read as impulse per second.

Conservation of momentum

When two objects interact, Newton's third law says they push on each other with equal and opposite forces, for exactly the same time. So they receive equal and opposite impulses, and whatever momentum one gains the other loses. Add them up and the total does not change.

Momentum is conserved in every collision and explosion of an isolated system, elastic or not; kinetic energy is not. In practice "isolated" means the external forces (friction, gravity balanced by normal forces) are small compared with the large internal forces during a brief collision. Gravity acting on a falling object is an external force, so a falling ball's momentum is not conserved; the ball plus Earth together conserve it.

Conservation applies to each direction separately. A clump of snow falling straight down into a moving sled adds mass but brings no horizontal momentum, so the sled slows: $m_{\text{sled}}v = (m_{\text{sled}} + m_{\text{snow}})v'$. The same is true of rain filling an open cart rolling along a level track. (The vertical momentum of the snow is absorbed by the ground through the normal force, so total momentum of sled and snow is not conserved in the vertical direction.)

Explosions and recoil

If a system starts at rest, its total momentum is zero, and stays zero. After an "explosion" (a gun firing, two skaters pushing apart, a compressed spring released between two carts, a person jumping off a cart), the pieces fly apart with equal and opposite momenta: $m_1 v_1 = -m_2 v_2$. The lighter piece moves faster. The kinetic energy comes from chemical energy, muscles or the spring, so KE increases in an explosion while momentum stays zero.

When you jump, you push Earth the other way. Its momentum change is exactly equal and opposite to yours, but its mass is about $6 \times 10^{24}$ kg, so its velocity change is immeasurably tiny. Even if every person on Earth jumped at once, Earth's speed would change by far less than the width of an atom per second.

Center of mass

The center of mass of a system is the mass-weighted average position of its parts.

So a rocket in deep space accelerates, but the center of mass of the rocket plus all the exhaust it has ejected does not. An exploding firework's fragments scatter, but their center of mass continues along the original parabola (until fragments hit the ground). This is also why the center of mass of a hovercraft and its load matters for balance (see "Where to put the load").

Collisions: elastic, inelastic and perfectly inelastic

In every collision of an isolated system, momentum is conserved. Collisions are classified by what happens to kinetic energy.

Total energy is always conserved; "inelastic" means kinetic energy is converted into other forms. Real collisions between steel balls, billiard balls or air-track gliders with spring bumpers come close to elastic. Collisions between atoms or molecules in a gas are treated as elastic in the kinetic theory of gases. A bullet lodging in a block, two train cars coupling, and clay hitting the floor are perfectly inelastic.

Perfectly inelastic collisions

When the objects stick, they share one final velocity, so one equation solves the problem.

Elastic collisions in one dimension

For an elastic head-on collision, solving momentum and kinetic-energy conservation together gives:

The special cases are worth memorizing:

Coefficient of restitution

The coefficient of restitution $e$ measures bounciness as the ratio of the speed of separation to the speed of approach.

Collisions in two dimensions

In two dimensions, momentum is conserved separately in $x$ and in $y$. Resolve every velocity into components, write two conservation equations, and solve. Total momentum before is a vector; after the collision, the vector sum of the pieces' momenta must equal it. A useful special case: when two equal masses collide elastically, off-center, and one starts at rest, they leave at 90° to each other.

When two objects collide at an angle and stick, the same component method gives the final velocity; then the KE lost is $KE_i - KE_f$.

Tests also run this backwards: they give the speeds before and after a sticking collision and ask for the angle between the two original paths. Square the momentum vectors. For masses $m_1, m_2$ with speeds $v_1, v_2$ at angle $\varphi$ to each other, $(m_1 + m_2)^2 v_f^2 = m_1^2 v_1^2 + m_2^2 v_2^2 + 2 m_1 m_2 v_1 v_2 \cos\varphi$. For two equal masses with equal speeds $v$, this simplifies to $v_f = v\cos(\varphi/2)$.

The ballistic pendulum

A classic two-stage problem. A bullet embeds in a hanging block (stage 1, perfectly inelastic: conserve momentum only). The block and bullet then swing up to height $h$ (stage 2: conserve mechanical energy). Never use energy conservation across stage 1.

Forces in a collision: the truck and the bug

A bug splatters on a truck's windshield. Which feels the larger force? Neither: Newton's third law makes the forces equal and opposite. Since they act for the same time, the impulses and the momentum changes are also equal in size. What differs is the acceleration and the change in velocity: the bug's tiny mass means a huge $\Delta v$; the truck's large mass means an imperceptible one. The same applies to a heavy and a light football player, a hammer and a nail, or your fist and a cinder block: the block pushes back on your hand as hard as you push on it, which is why it hurts.

Fluid statics: pressure, Pascal and Archimedes

What a fluid is

A fluid is a substance that flows and takes the shape of its container: liquids, gases and plasmas. The defining property is that a fluid at rest cannot sustain a shear stress. Push sideways on the top of a block of rubber and it bends and pushes back; push sideways on the surface of water and it simply flows. Liquids have a fixed volume and are nearly incompressible; gases expand to fill their container and are easily compressed.

Density and specific gravity

Density is mass per unit volume, $\rho = m/V$, in kg/m³ (or g/cm³; 1 g/cm³ = 1000 kg/m³). Specific gravity (relative density) is the ratio of a substance's density to that of water, so it has no units: ice has a specific gravity of about 0.92, mercury about 13.6.

Densities of liquids and gases change with temperature. Water is densest at about 4 °C (1000 kg/m³) and slightly less dense when warmer (998 kg/m³ at 20 °C). Because warm water is less dense, it gives a slightly smaller buoyant force on a fully submerged object. Ice is less dense than liquid water, which is why it floats.

Pressure and its units

Pressure is force per unit area, acting perpendicular to a surface.

The same force on a smaller area gives a larger pressure: a knife edge, a stiletto heel, a nail point. A larger area lowers the pressure: snowshoes, wide tank treads, and a hovercraft's cushion, which spreads the craft's weight over its whole footprint so its ground pressure is tiny (often less than a footprint's).

Two properties of pressure in a fluid at rest are tested directly:

1. It acts perpendicular to every surface it touches. A fluid at rest cannot push sideways along a surface (that would be a shear), so the force from pressure is always normal to the surface.
2. At a point, it is the same in all directions. A tiny cube anywhere in the fluid feels the same pressure on every face.

Pressure and depth

The fluid below any level holds up the weight of all the fluid above it. Take a column of liquid of height $h$ and area $A$: its weight is $\rho (Ah) g$, so the extra pressure at its base is $\rho g h$. The result is the hydrostatic equation.

Gauge pressure is the pressure above atmospheric, $P_{\text{gauge}} = P_{\text{abs}} - P_{\text{atm}} = \rho g h$. Tire gauges and blood-pressure cuffs read gauge pressure. Absolute pressure includes the atmosphere. A flat tire has a gauge pressure of zero but an absolute pressure of 1 atm.

The pressure depends only on depth, density and $g$, not on the shape of the container or the total amount of fluid. Vessels of very different shapes filled to the same depth have the same pressure at the bottom: the hydrostatic paradox. Connected vessels fill to the same level. A dam must be built to withstand the depth of the reservoir, not its length.

On a submerged cube, pressure is greatest on the bottom face (the deepest), least on the top, and in between on the sides. That pressure difference is the cause of the buoyant force (below).

Barometers and manometers

A barometer measures atmospheric pressure. Torricelli's mercury barometer is a tube closed at the top, filled with mercury and inverted in a dish. The mercury falls until the column's weight per area equals the atmosphere's push on the dish: $P_{\text{atm}} = \rho g h$, with a near-vacuum above the column. At sea level $h \approx 760$ mm. Any liquid works in principle; less dense liquids need taller columns.

The same reasoning explains why water stays in a straw when you cover the top with your finger and lift it out. Atmospheric pressure pushes up on the bottom of the column. The air trapped above expands very slightly, so its pressure drops a little below atmospheric, by exactly $\rho g h$ of the column, and the pressure difference holds the water up. A 10 cm water column needs the trapped air to sit only about 0.98 kPa (about 0.01 atm) below atmospheric. It is also why a suction pump cannot lift water more than about 10 m: the atmosphere can only push a water column that high.

A manometer is a U-shaped tube of liquid. One side connects to the gas being measured, and the other is open to the air. The difference in the liquid levels gives the gauge pressure: $P_{\text{gas}} - P_{\text{atm}} = \rho g \Delta h$. If the liquid is higher on the open side, the gas is above atmospheric pressure.

Air pressure and altitude

Air is a fluid too, so pressure falls as you climb. Over modest heights, where density barely changes, $\Delta P = \rho_{\text{air}} g\,\Delta h$ works: about 12 Pa per meter near sea level. Over kilometers, air thins as pressure drops, so pressure falls roughly exponentially, to about half its sea-level value near 5.5 km.

Pascal's principle and hydraulics

Pascal's principle: a change in pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of its container. A hydraulic lift uses it to multiply force. Push on a small piston of area $A_1$ with force $F_1$; the pressure $F_1/A_1$ reaches a large piston of area $A_2$, which pushes up with $F_2 = (F_1/A_1)A_2$. The pressure is the same on both pistons; the forces are not.

Car brakes, excavators, barber chairs and aircraft controls all use hydraulics. A hovercraft plenum is a loose cousin: the fan raises the pressure of the air in the plenum, and that pressure acts on the whole underside of the deck.

Archimedes' principle and buoyancy

Archimedes' principle: an object wholly or partly immersed in a fluid feels an upward buoyant force equal to the weight of the fluid it displaces.

Why it happens. Pressure increases with depth, so the fluid pushes up on the bottom of an object harder than it pushes down on the top. The sideways pushes cancel. The net upward push from that pressure difference is the buoyant force. Picture replacing the object with fluid of the same shape: that "fluid object" would float in balance, so the net pressure force must equal its weight. That is why the buoyant force depends on the volume displaced, not on what the object is made of. (Ultimately the pressure gradient exists because of gravity, so there is no buoyancy in free fall.)

Once an object is completely submerged, it displaces its full volume no matter how deep it goes, so (for a liquid of constant density) the buoyant force stays the same as it sinks deeper. The pressure on it grows; the difference between top and bottom does not.

Sink or float: compare densities

Whether a solid object sinks or floats depends on its average density compared with the fluid's, not on its weight alone. A fully submerged object has $W = \rho_{\text{obj}}Vg$ and $F_B = \rho_{\text{fluid}}Vg$, so:

A steel ship floats because its average density, steel plus all the air inside the hull, is less than water's. Load it with cargo of any kind (even light foam) and its weight rises, so it must displace more water and rides lower. A submarine changes its average density by flooding or emptying ballast tanks. "Heavy things sink and light things float" is wrong because volume matters: a 10-tonne log floats while a 1-gram pebble sinks.

Floating objects: the fraction submerged

A floating object is in equilibrium, so buoyant force equals weight: $\rho_{\text{fluid}}V_{\text{sub}}g = \rho_{\text{obj}}V g$. The fraction under the surface is simply the density ratio.

The same ratio runs backward: if a sphere floats with 50% submerged in water and 40% in another liquid, the liquid's density is $1000 \times 0.50/0.40 = 1250$ kg/m³.

Apparent weight, and finding density by weighing

Hang an object from a spring scale and lower it into water. The scale reads less, because the buoyant force carries part of the weight. The reading is the apparent weight.

The volume can also be measured directly by displacement: the rise in a graduated cylinder (in mL) is the volume in cm³. For a two-metal alloy of known total mass and volume, set up $m_1 + m_2 = m$ and $m_1/\rho_1 + m_2/\rho_2 = V$ and solve for the mass fractions.

Forces on a submerged object: normal force and tension

For an object resting on the bottom of a tank, three forces balance: weight down, buoyant force up, and the normal force from the floor up. So $N = W - F_B$, which is the apparent weight. For a floating object held fully under water by a string tied to the bottom, the string pulls down: $T = F_B - W$. For an object pushed under by a hand, the push is $F_B - W$.

Three classic puzzles

• Ice melting in a glass of water. A floating ice cube displaces exactly its own weight of water. When it melts, it becomes exactly that much water, which fills the hole it was displacing. The water level stays the same. (Floating sea ice melting does not raise sea level much; land ice sliding into the sea does.)
• A rock thrown overboard from a boat in a pool. In the boat, the rock displaces its weight of water (it is floating, via the boat), which is a large volume because rock is denser than water. On the bottom of the pool, it displaces only its own volume. The water level in the pool falls.
• A balloon in air. Air is a fluid, so it buoys up everything. A balloon rises when its total weight (gas plus skin plus load) is less than the weight of the air it displaces. Hot air and helium are less dense than the surrounding air. The net lift of a helium balloon is $(\rho_{\text{air}} - \rho_{\text{He}})Vg$ minus the weight of the balloon and payload. You also weigh slightly less on a scale in air than you would in a vacuum.

How a hovercraft differs from a boat

A boat floats by buoyancy: it sinks into the water until it displaces its own weight of water. A hovercraft is held up by cushion pressure: the fan keeps the air under the craft at a gauge pressure $p$ such that $p \times A_{\text{cushion}} = W$. Over water, the cushion does push a shallow dent into the surface, and the weight of water pushed aside by that dent equals the craft's weight, so the two ideas meet; but the hull itself stays out of the water, and over land or ice there is no buoyancy at all. The cushion pressure is small (a few hundred pascals for a model, a few kilopascals for a full-size craft) because the cushion area is large. The calculation and its consequences for the 2027 nickel payload are in Vehicle physics: floating on an air cushion.

Fluid dynamics: continuity and Bernoulli

The ideal fluid

The simple equations of fluid flow assume an ideal fluid:

• Incompressible: constant density (excellent for liquids; fine for air at speeds well below the speed of sound).
• Non-viscous (inviscid): no internal friction, so no energy is lost to heat.
• Steady: the velocity at each point does not change with time (laminar, not turbulent).
• Irrotational: fluid parcels do not spin about their own axes.

Real fluids break these assumptions; viscosity in narrow pipes and turbulence in fast flows are the usual culprits. One difference between an ideal fluid and a real one is simply that the ideal one has zero viscosity.

Flow rate and the continuity equation

The volume flow rate $Q$ is the volume passing a point per second. For flow at average speed $v$ through a cross-section of area $A$, $Q = Av$. The mass flow rate is $\rho Q = \rho A v$.

Because an incompressible fluid cannot pile up or vanish inside a pipe, whatever flows in one end must flow out the other at the same rate. That is the continuity equation, a statement of conservation of mass.

Where the pipe narrows, the fluid speeds up; where it widens, it slows down. Because area goes as radius squared, the speed changes with the square of the radius ratio: one third the radius means nine times the speed; going from the narrow end to the wide end, one ninth of the speed. Putting your thumb over a garden hose, or fitting a nozzle, works the same way.

Bernoulli's equation

Follow a small parcel of fluid along a streamline. If it speeds up, something must be pushing it forward: the pressure behind it must be higher than the pressure ahead. So in a speeding-up flow, pressure falls in the direction of flow. Bernoulli's equation makes this exact.

Derivation from the work-energy theorem. Consider a small volume $V$ of fluid moving from point 1 to point 2. The surrounding fluid does net work on it by pressure: $W = (P_1 - P_2)V$. By the work-energy theorem (itself a consequence of Newton's second law), that work equals the change in the parcel's kinetic plus gravitational potential energy: $(P_1 - P_2)V = \tfrac{1}{2}\rho V(v_2^2 - v_1^2) + \rho V g(h_2 - h_1)$. Divide by $V$ and rearrange, and you have Bernoulli's equation. So Bernoulli's equation is conservation of energy for a flowing fluid, applied per unit volume; its three terms are pressure (pressure energy), kinetic energy and potential energy.

Two special cases cover most problems:

• Same height: $P + \tfrac{1}{2}\rho v^2 = \text{constant}$. Where the fluid moves faster, its pressure is lower. This is Bernoulli's principle.
• Fluid at rest: $P + \rho g h = \text{constant}$, which is the hydrostatic equation $P = P_0 + \rho g h$ again.

Applications of Bernoulli's equation

Venturi meter

A venturi meter is a pipe with a smooth narrowing (the throat). By continuity the fluid speeds up in the throat, and by Bernoulli its pressure there drops. Measuring the pressure difference between the wide part and the throat gives the flow speed. The figure in Vehicle physics: floating on an air cushion shows one.

The same pressure drop drives carburetors, aspirators and perfume atomizers: fast air across the top of a tube lowers the pressure there, and the atmosphere pushes liquid up the tube.

Pitot tube and dynamic pressure

A pitot tube points into the flow. Fluid entering its open end is brought to rest (the stagnation point), so the pressure there is the static pressure plus the full dynamic pressure $\tfrac{1}{2}\rho v^2$. Comparing it with the static pressure measured by side holes gives the speed. Aircraft airspeed indicators work this way.

Torricelli's law: a hole in a tank

Water leaking from a hole a depth $h$ below the surface of a large open tank: at the surface, $P = P_{\text{atm}}$ and $v \approx 0$; at the hole, $P = P_{\text{atm}}$ too. Bernoulli leaves only kinetic and potential energy, which gives exactly the free-fall result.

A drain at the bottom of a full 100 m dam would, ideally, discharge at $\sqrt{2 \times 9.81 \times 100} = 44\ \text{m/s}$, and the gauge pressure there is $\rho g h \approx 9.8 \times 10^5$ Pa. Bernoulli also tells you how high a hose or fountain can throw water: a jet leaving at speed $v$ straight up rises to $v^2/(2g)$, the same as a ball.

Power carried by a flowing fluid

Pressure times volume is energy (Pa × m³ = J), so a pump that raises the pressure of a flow $Q$ by $\Delta P$ delivers power $P = \Delta P \cdot Q$. This is the same formula the guide uses for lift-fan power.

Lift, wings and other pressure effects

How a wing makes lift

A wing (airfoil) is shaped and tilted so that it turns the airflow downward. Two equivalent descriptions explain the lift:

• Newton's laws: the wing pushes air down, so the air pushes the wing up (third law). The rate of change of the air's downward momentum equals the lift.
• Bernoulli: the flow over the top of the wing is faster than the flow underneath, so the pressure on top is lower; the pressure difference times the wing area is the lift.

Both are correct and describe the same flow; neither is the "real" one. What is wrong is the popular equal transit time explanation, which claims air over the longer top surface must "catch up" with the air underneath at the trailing edge, and so must go faster. Nothing requires the two parcels to meet; in fact the air over the top arrives at the trailing edge well before the air underneath, and the speed difference is much larger than the path-length difference would predict. Flat or symmetric wings, and planes flying upside down, make lift too, which equal transit time cannot explain. (NASA Glenn's pages on incorrect lift theories discuss this.)

Angle of attack is the angle between the wing's chord line and the oncoming air. Lift increases roughly in proportion to the angle of attack, up to a critical angle, typically around 15° for a common airfoil (the exact value depends on the airfoil shape). Beyond it, the flow separates from the upper surface, lift drops sharply and drag rises: the wing stalls. A test asking which angle gives the most lift wants the largest angle that is still below the stall.

Roofs, curtains and spinning balls

• Roofs in a storm. Wind blowing fast over a roof lowers the pressure above it, while the air inside the house is nearly still at normal pressure. The net force pushes the roof up. The size of the effect is about $\tfrac{1}{2}\rho v^2$ times the roof area. Opening windows on the downwind side helps equalize the pressure.
• The shower curtain that bulges inward is often explained by Bernoulli (moving air and water inside); the full explanation also involves the swirling air the spray sets up and warm air rising, so treat it as a qualitative example.
• Curveballs and the Magnus effect. A spinning ball drags air around with it, which deflects the flow behind the ball to one side; the air pushes the ball the other way. Topspin makes a tennis ball dip; backspin lifts a golf ball; sidespin curves a baseball or soccer ball.
• Two ships or trucks passing close together are pushed toward each other by the faster, lower-pressure flow between them.

Bernoulli and the hovercraft

Under a hovercraft, the plenum air is nearly still at the cushion pressure; at the skirt gap it accelerates out to the surrounding air. Bernoulli converts the cushion's gauge pressure into exit speed, $v = \sqrt{2\Delta p/\rho}$, which sets how much air the fan must supply. That calculation is in "How much harder the fan works with a full load". Bernoulli is also why the pressure in a fast-moving stream (the fan's outflow) is not the same as the pressure in the slowed, trapped cushion beneath the deck.

How real hovercraft work: history and design

Ways to hold up a cushion

Every air cushion vehicle solves the same problem: keep a volume of air under the hull at a pressure a little above atmospheric, while that air constantly leaks out around the edges. The three classic designs differ in how they slow the leak.

• Plenum chamber craft. The simplest design: a fan blows air into a box that is open at the bottom (the plenum). Pressure builds until it lifts the craft; air then leaks out under the edges. It works, but with no seal the leak is large, so it needs a lot of fan power for a small hover height. Thornycroft's and most early ideas were of this kind, and many simple model hovercraft are too.
• Momentum curtain (peripheral jet) craft. Cockerell's insight. Instead of filling the whole underside, air is blown down and inward through a narrow slot running around the edge of the hull. This high-speed sheet of air (the air curtain) has momentum that resists being pushed outward by the cushion, so it acts as a moving wall that traps a higher-pressure cushion inside for the same air supply. The SR.N1 used this principle and had no skirt at first.
• Skirted craft. A flexible skirt of rubberized fabric hangs below the hull and holds the cushion in. Because the skirt can deflect over waves and obstacles and spring back, the hull can ride high above the surface while the actual leak gap under the skirt (the hover gap) stays small. This multiplies the obstacle clearance for the same fan power, and it is the design almost all modern hovercraft use, from rescue craft to the LCAC.

A related family, the sidewall hovercraft or surface effect ship, has rigid side hulls that dip into the water and flexible seals only at bow and stern.

Compared with a fully skirted hovercraft, a surface effect ship trades amphibious ability for performance on open water. Its rigid side hulls seal the cushion along both sides, so far less air leaks out and the lift fans need less power. The hulls also sit in the water, so it can be driven by water propellers or waterjets, which push more efficiently on water than air propellers push on air, and the hulls resist the sideways wind drift that makes a hovercraft hard to hold on course. The result is generally better seakeeping and higher sustained speed in rough water. The price: it cannot cross beaches, mud, ice or land, which is the main reason to build a hovercraft at all.

Skirt types

The early history of the skirt runs through Britain. C. H. Latimer-Needham suggested a flexible skirt in 1958, building on an idea from his business partner Arthur Ord-Hume, and sold his skirt patents to Westland in 1961. At Hovercraft Development Ltd, Denys Bliss improved the design by replacing two separate sheets with a single sheet bent into a U shape, with slots cut in the bottom to form the vents. (Some older test keys attribute this U-shaped slotted skirt to Charles Fletcher; the sources found for this guide credit Bliss, so check the key's source if you meet that question.) Skirts are made of tough, flexible fabric, usually nylon coated with rubber or neoprene. Skirt wear is a major running cost for full-size craft.

An even air supply around the skirt matters as much as its shape: if one side gets less air, that side sinks, rubs and drags, and the craft veers toward it. In a model hovercraft, smooth rounded corners and a plenum that distributes air evenly reduce turbulence and losses.

Lift, thrust and steering

Lift and thrust. Small craft often use one engine or fan for lift and another for thrust (two engines; extra engines add thrust or redundancy). An integrated design uses a single fan and splits its output: part goes down into the cushion, part goes back through a thrust duct. The 2027 event allows either, since all lift and thrust must come from air.

Steering. With almost no friction against the ground, a hovercraft cannot steer the way a car does, by pushing sideways on the road. Instead it redirects air:

• Rudders in the propeller slipstream deflect the thrust air sideways.
• Swivelling propellers or pylons point the thrust (the SR.N4's four propellers were mounted on pylons that turned).
• Puff ports (bow thrusters) let out jets of cushion air sideways to push the bow around at low speed.
• Differential thrust from two propellers, and on small racing craft, shifting the driver's weight, which tilts the cushion and lets air escape more on one side.

Why hovercraft are hard to control. Low friction is the whole point, but it cuts both ways. A hovercraft keeps sliding in its old direction when it turns (inertia, Newton's first law), so it drifts sideways through a curve. It cannot brake against the ground; it must reverse its thrust, swing its propellers, or settle onto the surface to stop. A crosswind or a slope pushes it sideways with nothing to resist. That is why hovercraft are said to be harder to control than cars: they lack friction with the surface.

Ground effect is not air cushion. A wing-in-ground-effect (WIG) craft, such as the Soviet ekranoplans developed by Rostislav Alekseyev (the huge "KM", nicknamed the "Caspian Sea Monster", first flew in 1966), flies on wings just above the water and gets extra lift from the air squeezed between wing and surface at speed. It needs forward speed to stay up. A hovercraft's cushion is made by fans and works even when the craft is standing still.

History of the hovercraft

The air cushion idea is old; making it practical needed light, powerful engines and a way to hold the cushion in.

Hovercraft at work

Military. The first air cushion craft built for a military purpose was Müller von Thomamühl's 1915 torpedo boat. Since the 1960s, militaries have used hovercraft mainly as amphibious landing craft and patrol craft, because they can cross beaches, mud flats, marshes, ice and shallows that stop conventional boats. The US Navy's LCAC (Landing Craft Air Cushion) carries about 60 short tons (75 overloaded) at more than 40 knots from a ship's well deck straight onto the beach, crewed by five sailors. It is being replaced by the Ship-to-Shore Connector (LCAC 100 class), whose first craft were delivered around 2020. The Soviet Union built the largest hovercraft fleet, including the Aist and the giant Zubr class, which can carry three main battle tanks or several hundred troops.

Civilian. Passenger ferries (the SR.N4 across the Channel, Hovertravel across the Solent), search and rescue over ice, mud flats and floodwater where boats ground and people cannot walk, coast guard work on frozen rivers and lakes, survey and exploration in marshes and the Arctic, and recreation and racing. Records include the French N500 Naviplane's reported 74-knot (137 km/h) average on a Boulogne–Dover run and the British experimental tracked hovercraft RTV 31, which reached about 167 km/h (104 mph) on a guideway on 7 February 1973.

Advantages and disadvantages

What the 2027 event rules mean for steering

The 2027 event has no written test, but students are judged on questions about their vehicle's design and operation. Two rules shape what a model craft can do. Electronics are limited to simple parts with no integrated circuits (other than those built into a commercial motor), and once your hands leave the vehicle at the start you may not touch or interfere with it until the Event Supervisor says so. Together, these mean that nothing can steer the craft during a run: there is no radio link, and no one may guide it by hand. It must run straight on its own, with the track's side rails as the only guide. That puts all the weight on the ideas in this chapter: an even air supply around the skirt, a centered load, and symmetric thrust, so the craft does not yaw into a rail. See "Construction rules and why they exist" for the exact rule summary.

Common pitfalls

How tournaments test this

Hovercraft in 2027 has no written test. Your score comes entirely from your runs: the Distance, Time and Mass Scores of your best run, adjusted by penalty multipliers. The Event Supervisor also checks the vehicle against the construction and safety rules before your competition time and may ask either partner questions about how the vehicle was designed, built and operated. Being unable to explain your own vehicle is a problem under the building policy, so both partners should be able to describe the lift system, the thrust and speed control, the payload holder and the calibration method.

The scio.ly question bank still holds the written Hovercraft tests from seasons when the event had one. Of the 951 questions tagged to this event, about 87% come from the 2016-17 and 2017-18 seasons and about two thirds from Division C tests; 35% are multiple choice and the rest are short free-response or calculation items. They are a full introductory mechanics and fluids test with a hovercraft in many of the stories, which is why this guide teaches that physics in full. The share of questions on each area, and where this guide answers it:

Roughly four questions in ten need a calculation, so work every example in the physics chapters with a calculator in hand. The rest are definitions, true/false misconceptions and "which law explains this" items; the trap tables at the end of each chapter target exactly those. A few archive keys are wrong, especially in history trivia; where a key disagrees with the sources, the history chapter says so.

Test yourself

These archived tournament questions cover the physics behind a loaded hovercraft. They come from seasons with a written test and are practice for understanding and explaining your vehicle, not a preview of a 2027 test. Answer each before you open the link, and check the official key against the explanation.

Test yourselfTest yourself: lift, cushion pressure and payload

0 / 0 correct

1. 1

   The “Renegade IQ Hovercraft” has a mass of 161 kg. How much downward thrust in kN is required to make it hover when there are two 75.0 kg passengers on board?

   Show answer

   From: Troy Invitational, 2017-18, Division C

1 question from this set is no longer available.

Test yourselfTest yourself: thrust, motion and stopping

0 / 0 correct

1. 1

   A 750 kg hovercraft is traveling at a constant speed while its fan generates a continual thrust of 1500 Newtons. What is the sum of all the drag forces on the craft.

   Show answer

   From: Columbia Invitational, 2017-18, Division C

2 questions from this set are no longer available.

The sets below are drawn from the same archive and cover the full physics test: forces, motion, energy and momentum, fluids, and hovercraft design and history. Each line links to its source test; the explanation shows the method.

Test yourselfTest yourself: forces and Newton's laws

0 / 0 correct

1. 1

   If two people are pushing against each other, along the same line with a force of 400 N and 450 N, there is a/an:

   Question 1 options

   ABalanced ForceBUnbalanced ForceCWeak ForceDStrong Force

   From: Westlake Invitational 2018, Division B

Test yourselfTest yourself: kinematics and projectiles

0 / 0 correct

1. 1

   A hovercraft has the initial velocity of 20m/s. How long must it accelerate at a constant rate of 5 m/s² before its average velocity is equal to twice its initial velocity?

   Question 1 options

   A5 sB10 sC8 sD30 s

   From: Sylvania Northview Invitational 2016, Division C

Test yourselfTest yourself: energy, momentum and collisions

0 / 0 correct

1. 1

   A 1-kg mass has a kinetic energy of 1 Joule when its speed is

   Question 1 options

   A0.45 m/sB1 m/sC1.4 m/sD4.4 m/sE0.5 m/s

   From: Mentor Science Olympiad 2018, Division B

Test yourselfTest yourself: fluids and pressure

0 / 0 correct

1. 1

   Forces exerted by a static fluid on an object are always perpendicular to the surfaces of the given object

   Question 1 options

   ATrueBFalse

   From: Yale Invitational 2017-18, Division C

Test yourselfTest yourself: hovercraft design and history

0 / 0 correct

1. 1

   The first recorded design for an air-cushioned vehicle was designed in 1716. Who designed it?

   Question 1 options

   AChristopher CockerellBEmanuel SwedenborgCDagobert MullerDBenjamin Franklin

   From: Troy Invitational 2017, Division C

Condensed facts

A condensed sheet of the formulas, constants and facts from this guide, for printing and for your impounded notes. Every symbol is defined in the chapter it comes from.

References and further reading

• Science Olympiad: the 2027 rules manual (Hovercraft B, pages B31 to B34; Hovercraft C, pages C37 to C40), the building policy, example track and ramp setups, and the scoring spreadsheet. The manual is the authority; this guide paraphrases it.
• U.S. Mint, coin specifications: mass, diameter and thickness of the nickel.
• SciOly Wiki: Hovercraft: the community page for the event. Link only; this guide does not reproduce its content.
• OpenStax, University Physics Volume 1: Newton's laws, impulse, kinetic energy, friction and fluid pressure.
• NASA Glenn Research Center, Bernoulli's principle: pressure and speed in a moving fluid.
• OpenStax, College Physics 2e: algebra-based kinematics, projectile motion, Newton's laws, momentum, energy, friction and drag, and fluid statics and dynamics at the level of the archived tests.
• HyperPhysics (Georgia State University): concise concept maps for mechanics and fluids, useful for checking a formula quickly.
• NASA Glenn Research Center, Beginner's Guide to Aeronautics: lift, drag, the drag equation and why the equal-transit-time explanation of lift is wrong.
• The Hovercraft Museum: history of Cockerell, the SR.N1 and later craft.
• Figure credits appear beside each figure. All figures are from Wikimedia Commons under free licenses.