Work, Energy & Momentum — Print · Physics & Environmental Science, Grade 12

Work & Power

Learning goals

• I can calculate the work done by a force using W = Fd (and W = Fd·cosθ when the force acts at an angle).
• I can apply the work-energy theorem and calculate power with P = W/t.

Readings for this unit

• A slower cart has not destroyed its energy

In everyday speech 'work' means almost any effort, but in physics it has a sharp, specific meaning: work is the energy a force transfers to an object as that object moves. When a force pushes something through a displacement in the same direction, the work done is simply W = Fd — force times distance. Because a joule is a newton-metre, pushing with 10 N through 3 m transfers 30 J of energy. Work is the bridge between force (Unit 2) and energy: a force by itself does nothing to an object's energy until the object actually moves.

Direction matters, which is why the full formula is W = Fd·cosθ, where θ is the angle between the force and the displacement. When you carry a heavy backpack horizontally across a room, gravity pulls straight down while you move sideways — the angle is 90°, cos 90° = 0, and gravity does zero work on the pack even though holding it feels exhausting. When friction drags backward against motion, the angle is 180°, cos 180° = −1, and friction does negative work, removing energy from the object. Only the part of a force that lies along the motion actually transfers energy.

The payoff of tracking work is the work-energy theorem: the *net* work done on an object equals its change in kinetic energy, W_net = ΔKE. Push a cart and the positive net work speeds it up; let friction act and the negative net work slows it down. This theorem is a shortcut of real power — if you know the net work, you know exactly how much the object's motion energy changed, without having to follow the messy details of acceleration and time in between.

Doing work is one thing; how *fast* you do it is another, and that is power. Power is the rate of doing work, P = W/t, measured in watts (one watt is one joule per second). Two cranes might lift identical loads to the same rooftop — doing exactly the same amount of work — yet the crane that finishes in half the time delivers twice the power. Power can also be written P = Fv, the product of force and speed, which is why a car engine must supply more power to hold a high speed against air resistance than to cruise slowly.

These ideas quietly rate almost every machine around you. The 'horsepower' on an engine, the wattage on a motor or a light bulb, the output of a power plant — all are statements about how quickly energy is being transferred. Engineers care about both totals and rates: a small motor and a huge one might eventually do the same job, but only the powerful one can do it fast enough to matter. Master work and power and you hold the vocabulary for describing every energy transfer in the rest of this unit.

Model check: A force perpendicular to displacement does no mechanical work in the point-particle model. W=30 J; P=W/t=15W, if that force is constant.

Work

W = Fd·cosθ

F is the force in newtons (N); d is the displacement in metres; θ is the angle between the force and the displacement; W is in joules (J). When the force is along the motion, cosθ = 1 and W = Fd.

Work-Energy Theorem

W_net = ΔKE = ½mv_f² − ½mv_i²

the net work done on an object equals its change in kinetic energy; v_i and v_f are the initial and final speeds.

Power

P = W / t (also P = Fv)

W is work in joules; t is time in seconds; P is power in watts (1 W = 1 J/s); v is speed in m/s.

Worked examples

Avoid these mistakes

• A force perpendicular to displacement does no mechanical work in the point-particle model.

Words to know

work

The energy transferred to an object when a force acts on it through a displacement: W = Fd·cosθ, measured in joules (J).

joule (J)

The SI unit of work and energy; one joule is one newton-metre (1 J = 1 N·m).

work-energy theorem

The net work done on an object equals its change in kinetic energy: W_net = ΔKE.

power

The rate at which work is done or energy is transferred: P = W/t, measured in watts (W).

watt (W)

The SI unit of power; one watt is one joule per second (1 W = 1 J/s).

James Watt and the invention of 'horsepower'

When Scottish engineer James Watt was selling his improved steam engines in the 1770 s, customers wanted to know how many horses each machine could replace. So Watt measured how much work a strong draft horse could do — hauling loads out of a mine — and defined one horsepower as about 550 foot-pounds per second, which works out to roughly 746 watts. It was brilliant marketing dressed as physics, and it stuck: car engines are still rated in horsepower today, even though the SI unit of power, the watt, is named after the very same man.

Misconception: holding a heavy weight is 'hard work' but does no physics work

Stand holding a heavy box perfectly still and your arms will soon burn — yet in the language of physics you are doing zero work on the box. Work is W = Fd, and if the box does not move, d = 0, so no energy is transferred to it. Why does it feel so tiring? Your muscle fibres are rapidly twitching and relaxing to hold the tension, doing internal biological work and generating heat inside you, but none of that energy goes into the box. It is a sharp reminder that the physics definition of work is about energy transferred to the object, not about human effort.

Negative work and the angle in W = Fd·cosθ

The cosine in W = Fd·cosθ is not a technicality — it decides the sign of the work. When a force points partly against the motion, cos θ is negative and the force removes energy: friction sliding a box to a stop (θ = 180°, cos = −1) does negative work, draining kinetic energy into heat. When a force is exactly perpendicular to the motion (θ = 90°, cos = 0), it does no work at all — which is why the tension in a string does zero work on a ball whirling in a circle, and why gravity does no work on a satellite in a circular orbit even though it is always pulling. Only the component of force along the displacement transfers energy.

Real-world power: from a light bulb to a rocket

Power ratings span an astonishing range. A human can sustain about 100 W of useful mechanical output for hours (roughly the power of an old light bulb), while a trained sprinter can hit over 1,000 W for a few explosive seconds. A family car's engine peaks near 100,000 W (about 130 hp), and the first stage of a Saturn V rocket delivered on the order of 10¹¹ W at liftoff. Since all of these do work by transferring energy, they can be compared on a single scale of watts — the same unit whether you are rating a nightlight or a moon rocket.

In the real world

Engineers rate every motor, engine, and power plant by its power — how fast it can transfer energy — because a job done too slowly (charging a car, pumping floodwater, lifting an elevator) can be as useless as not doing it at all.

Exit check

A10N parallel force moves a box 3 m in 2 s. Find work and average power.

Kinetic & Potential Energy

Learning goals

• I can calculate kinetic energy with KE = ½mv² and gravitational potential energy with PE = mgh.
• I can calculate elastic potential energy stored in a spring using PE = ½kx².

Readings for this unit

• A slower cart has not destroyed its energy

The energy of a moving object is its kinetic energy, KE = ½mv². The most important feature of this formula is that speed is *squared*, so kinetic energy is exquisitely sensitive to how fast something moves. Double an object's speed and its kinetic energy does not double — it quadruples; triple the speed and it grows nine-fold. This single fact explains why a highway crash releases so much more energy than a parking-lot fender-bender, and why even a light object, like a hailstone or a fast-pitched ball, can strike with surprising force when its speed is high.

Gravitational potential energy is the energy an object stores simply by being up high, PE = mgh, where m is mass, g = 9.8 m/s² near Earth's surface, and h is height above some chosen reference level. Raise a mass higher, or choose a heavier mass, and you bank more energy that gravity can later cash in as motion. A subtle but important point: the value of h depends on where you decide to call 'zero.' Only *changes* in height — and therefore changes in potential energy — carry physical meaning, so you are free to place the reference wherever makes a problem simplest.

Springs and other elastic objects store a third kind: elastic potential energy, PE = ½kx², where k is the spring constant (the spring's stiffness) and x is how far it is stretched or compressed from its natural length. This comes straight from Hooke's law, F = kx: because the restoring force grows steadily as you stretch a spring, the energy stored is the area under that force-versus-stretch line — a triangle, which is where the factor of ½ comes from. Compress a spring twice as far and you store four times the energy, ready to launch a dart, a pinball, or a valve.

These forms are not isolated curiosities; they are constantly converting into one another. A ball at the top of its arc holds mostly gravitational potential energy and almost no kinetic energy, then trades one for the other as it falls. A drawn bow stores elastic potential energy that becomes the arrow's kinetic energy at release. Tracking how much energy lives in each form at each moment — motion energy versus stored positional energy — is exactly the modelling skill this unit is building toward, and it sets up the conservation law of the next lesson.

Because all three are measured in the same unit, the joule, you can add them, compare them, and follow them as a single running total. That is what makes energy such a powerful accounting tool: an engineer can look at a roller-coaster car, a compressed spring in a mattress, or water held behind a dam and put a single number — in joules — on the energy each holds, then predict what that energy will do when it is released.

Model check: Gravitational potential energy needs a chosen reference; negative values can be valid. U=mgh=58.8 J; changing the reference shifts absolute U but not energy differences.

Kinetic Energy

KE = ½mv²

m is mass in kg; v is speed in m/s; KE is in joules (J). Speed is squared, so KE grows fast as v increases.

Gravitational Potential Energy

PE = mgh

m is mass in kg; g = 9.8 m/s² near Earth's surface; h is height above the reference level in metres; PE is in joules (J).

Elastic Potential Energy

PE = ½kx²

k is the spring constant in N/m; x is the stretch or compression from the natural length in metres; PE is in joules (J).

Worked examples

Avoid these mistakes

• Gravitational potential energy needs a chosen reference; negative values can be valid.

Words to know

kinetic energy

The energy an object has because of its motion: KE = ½mv², measured in joules (J).

gravitational potential energy

Stored energy an object has because of its height in a gravitational field: PE = mgh.

elastic potential energy

Energy stored in a stretched or compressed elastic object such as a spring: PE = ½kx².

spring constant (k)

A measure of a spring's stiffness, in newtons per metre (N/m); it appears in Hooke's law F = kx.

reference level

The chosen height where gravitational potential energy is defined as zero; only changes in PE have physical meaning.

The 'vis viva' controversy: was motion measured by mv or mv²?

In the early 1700 s, Europe's best thinkers fought over how to quantify a moving object's 'force.' Followers of Descartes backed quantity of motion, mv (what we now call momentum); Leibniz argued for vis viva, or 'living force,' proportional to mv². Experiments by Willem 's Gravesande — dropping brass balls into soft clay and measuring the depth of the dent — showed the effect grew with the square of the speed, and the mathematician Émilie du Châtelet helped make the case decisively. The resolution was that both quantities are real and useful: mv is conserved as momentum, and ½mv² is the kinetic energy. This whole unit is, in a sense, the happy ending to that 300-year-old argument.

Misconception: potential energy is not an absolute number

It is tempting to think an object 'has' a definite amount of gravitational potential energy, but PE = mgh depends entirely on where you set h = 0. A book on a second-floor desk has one PE value measured from the floor, a larger one measured from the ground outside, and a negative one measured from the ceiling. None is more 'correct' than the others. What is physically real — and always comes out the same — is the change in PE as the object rises or falls. Choosing a convenient reference level is a freedom that often makes a hard problem easy.

Where the ½ in ½kx² comes from: the area under Hooke's law

A spring obeys Hooke's law, F = kx: the more you stretch it, the harder it pulls back, in a straight-line relationship. To find the energy stored you cannot just multiply force by distance, because the force keeps changing as you stretch. Instead you take the area under the force-versus-stretch graph — and since that graph is a straight line from 0 up to kx, the area is a triangle: ½ × base × height = ½ × x × (kx) = ½kx². That same 'area under a changing force' reasoning is why kinetic energy also carries a ½, and it is your first taste of the calculus that underlies all of energy physics.

Storing energy at grid scale: pumped hydro and giant springs

The formulas in this lesson run some of the world's largest machines. Pumped-storage plants use surplus night-time electricity to pump enormous volumes of water uphill, banking it as gravitational PE = mgh; at peak demand they let it fall back through turbines, recovering about 80% of the energy. Elastic potential energy is stored at scale too — from the coiled springs in mechanical watches to experimental flywheel and spring systems that smooth out power grids. Whenever engineers need to save energy now and release it later, they reach for exactly the potential-energy equations you are practicing here.

In the real world

Pumped-storage power stations bank surplus electricity by pumping water uphill into a reservoir as gravitational potential energy, then release it through turbines at peak demand — a giant rechargeable battery built out of PE = mgh.

Exit check

A2 kg mass at 3 m has what U usingg=9.8 andU=0 atground?

Conservation of Energy

Learning goals

• I can apply conservation of mechanical energy (KE + PE = constant) to solve for speeds and heights in ideal systems.
• I can account for energy dissipated by friction and calculate the efficiency of a real device.

Readings for this unit

• A slower cart has not destroyed its energy

Mechanical energy includes kinetic and appropriate potential-energy stores for a selected system. It can remain constant when external transfers and nonconservative effects are negligible in the stated account. Absence of friction alone is not sufficient if an external motor does work or energy leaves by another pathway. For an ideal loss-free descent, gravitational potential energy can transfer into kinetic energy while their sum stays constant.

This is not just elegant; it is a ruthlessly efficient problem-solving tool. Suppose a ball is dropped from a height h and you want its speed at the bottom. Setting mgh = ½mv², the mass cancels and you get v = √(2gh) — the answer in a single line, with no need to track acceleration or time at all. Conservation lets you leap straight from a starting condition to a final one, ignoring the complicated journey in between, which is why physicists reach for it constantly.

Friction and other interactions can reduce translational mechanical energy while increasing thermal, sound, deformation, rotational, or other stores. A complete energy account includes transfers across the selected boundary. For an isolated system, total energy is conserved; for an open energy account, initial plus incoming equals final plus outgoing. An unexplained residual identifies missing information or a model limitation rather than proving energy has been destroyed.

Efficiency is useful output divided by input for a specified service and boundary. A motor receiving 800 J and delivering 600 J useful work is 75% efficient; account for the remaining 200 J as other transfers or storage rather than assuming its exact destination without evidence. Thermal engines have a second-law limit under their operating conditions, but it is too broad to say every kind of defined efficiency must always be below 100%. An ideal electrical heater can convert electrical input to thermal output with a near-unity device account; a heat pump’s coefficient of performance is a different ratio that includes transferred environmental heat.

A Newton's cradle captures the whole story in one desktop toy. Lift a steel ball and release it, and the energy races through the row and out the far side, swing after swing, almost perfectly conserved. Almost — each click you hear is a sliver of energy leaving as sound, and each collision warms the steel a touch, so the swings slowly shrink and finally stop. Nothing was destroyed; the energy merely dissipated. The same accounting lets engineers design hydroelectric dams, regenerative brakes, and power plants, all of which live or die by how much of their energy they can steer into a useful form instead of losing it to heat.

Learning to write down every joule at the start, track where each one goes, and insist that the books balance at the end is the heart of energy analysis. Whether the 'missing' energy went into motion, into height, into a stretched spring, or into scattered heat, conservation guarantees it went *somewhere* — and that certainty is what turns a tangle of forces into a solvable problem.

Model check: Mechanical energy can decrease while total energy remains conserved. Energy transfers into thermal energy, deformation, or sound of the chosen larger system; account for boundaries.

Mechanical Energy

E = KE + PE

the sum of an object's kinetic and potential energy.

Conservation of Mechanical Energy

KE_i + PE_i = KE_f + PE_f

valid when friction and other non-conservative forces are absent; the total of kinetic plus potential energy stays constant.

Efficiency

efficiency = (useful output ÷ total input) × 100%

the percentage of input energy that emerges as useful output; always less than 100% for a real device.

Worked examples

Avoid these mistakes

• Mechanical energy can decrease while total energy remains conserved.

Words to know

mechanical energy

The sum of an object's kinetic and potential energy: E = KE + PE.

conservation of mechanical energy

In the absence of friction and other non-conservative forces, KE + PE stays constant: KE_i + PE_i = KE_f + PE_f.

law of conservation of energy

The total energy of an isolated system never changes; energy is only transformed or transferred, never created or destroyed.

dissipation

The spreading of useful mechanical energy into scattered thermal energy and sound, usually through friction or drag.

efficiency

The fraction of input energy that comes out as useful output: efficiency = (useful output ÷ total input) × 100%.

Joule's paddle wheel: proving heat is energy

In the 1840 s, English brewer and physicist James Prescott Joule ran a famous experiment: a falling weight turned a paddle wheel inside an insulated barrel of water, and Joule measured the tiny temperature rise as the paddles churned. He found a fixed exchange rate between mechanical energy lost and heat gained — the 'mechanical equivalent of heat.' This showed that the energy 'lost' to friction was not gone but had become thermal energy, and it was decisive evidence for the law of conservation of energy. Fittingly, the SI unit of energy, the joule, is named after him.

Why we can't recycle waste heat: the second law

Conservation of energy guarantees the total never changes, which raises a puzzle: if the heat wasted by an engine is still energy, why can't we just collect it and reuse it to get 100% efficiency? The answer is the second law of thermodynamics: energy spontaneously spreads from concentrated, organized forms into disordered, scattered heat, and reversing that spread perfectly is impossible. This is why every perpetual motion machine ever built has failed, and why patent offices now refuse to even consider them. Energy is always conserved (first law), but its usefulness steadily degrades (second law).

Conservative vs non-conservative forces

Mechanical energy is only conserved when the forces at work are conservative — forces like gravity and ideal springs, for which the energy depends only on position, not on the path taken. Lift a book straight up or along a winding ramp to the same shelf, and gravity stores the same PE either way. Friction and air resistance are non-conservative: they always oppose motion, drain energy along the whole path, and turn it into heat that cannot be recovered as motion. Recognizing which forces are present tells you instantly whether you can use the clean 'KE + PE = constant' shortcut or must also subtract the energy dissipated.

Regenerative braking: conservation of energy on the road

An ordinary car brake throws away kinetic energy as useless heat in the brake pads — you can smell it after a long downhill. Electric and hybrid vehicles do something cleverer with regenerative braking: the electric motor runs backward as a generator, using the car's kinetic energy to recharge the battery instead of just heating the brakes. It is conservation of energy turned into engineering profit, recovering a large fraction of the energy that would otherwise be lost and noticeably extending an EV's range, especially in stop-and-go city driving.

In the real world

Electric and hybrid cars use regenerative braking to capture the kinetic energy that would normally be wasted as brake heat, converting it back into stored electrical energy in the battery — a direct application of conservation of energy that noticeably extends driving range.

Exit check

A cart loses 20 J mechanical energy due to friction. Where is it?

Momentum & Collisions

Learning goals

• I can calculate momentum with p = mv and apply the impulse-momentum relationship J = FΔt = Δp.
• I can use conservation of momentum to analyze elastic and inelastic collisions.

Readings for this unit

• A slower cart has not destroyed its energy

Momentum is an object's 'quantity of motion,' defined simply as p = mv — mass times velocity. Because it multiplies the two, a slow but massive object and a fast but light one can carry the same momentum: a loaded truck rolling at walking pace and a bullet in flight are both hard to stop. Momentum is a *vector*, so direction counts; something moving left has momentum opposite in sign to something moving right, and that bookkeeping of signs is what makes momentum such a precise tool. Its unit is the kilogram-metre per second (kg·m/s).

To change an object's momentum you must apply a force for some time, and that combination is called impulse: J = FΔt. The impulse delivered exactly equals the change in momentum, J = Δp — the impulse-momentum theorem. This relationship has a life-saving twist. The momentum change in a car crash is fixed by how fast you were going, but F = Δp/Δt shows that stretching the collision over a longer time Δt shrinks the force. That is precisely how airbags, crumple zones, padded dashboards, and even bending your knees when you land all protect you: they lengthen the stopping time and soften the force.

The crown jewel of this lesson is conservation of momentum: if no net external force acts on a system, its total momentum is identical before and after any interaction. This follows straight from Newton's third law — in a collision the two objects push on each other with equal and opposite forces for the same time, so the impulses cancel and whatever momentum one gains, the other loses. It holds whether objects bounce apart, stick together, or explode outward, which is why it works for colliding billiard balls, coupling train cars, and a rocket hurling exhaust one way to drive itself the other.

Collisions come in two flavours, and telling them apart hinges on energy. In an elastic collision, both momentum *and* kinetic energy are conserved; the objects bounce off with no energy lost to heat, the way hardened steel balls or gas molecules nearly do. In an inelastic collision, momentum is still conserved but some kinetic energy is converted into heat, sound, and permanent deformation. The extreme case is a perfectly inelastic collision, in which the objects stick and move off together — like a dart burying itself in a board — losing the most kinetic energy possible while still, always, conserving momentum.

A neat special case makes the ideas concrete: in a head-on elastic collision between two *equal* masses, they simply exchange velocities. A moving ball striking an identical stationary one stops dead while the target shoots off at the incoming speed — exactly what you see click across a Newton's cradle. Analyzing any collision then becomes a two-part checklist: momentum is *always* conserved, so write p_before = p_after; kinetic energy is conserved *only if* the collision is elastic. Keeping those two rules straight lets you predict the outcome of impacts from the subatomic to the astronomical.

Momentum conservation is one of the deepest principles in physics, holding even where Newton's laws break down, in the quantum and relativistic worlds. For everyday engineering it is indispensable: it tells crash-safety teams how to design a survivable impact, lets rocket scientists compute thrust in the vacuum of space where there is nothing to push against, and allows physicists to reconstruct particle collisions they never directly see — all by insisting that the total momentum going in must equal the total coming out.

Model check: Momentum conservation does not automatically imply kinetic-energy conservation. Initialp=6 kg·m/s, totalm=3 kg, so vf=2 m/s with negligible external impulse.

Momentum

p = mv

m is mass in kg; v is velocity in m/s; p is a vector in kilogram-metres per second (kg·m/s).

Impulse-Momentum Theorem

J = FΔt = Δp

F is the average force; Δt is the contact time; the impulse equals the change in momentum, Δp = mv_f − mv_i.

Conservation of Momentum

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

with no net external force, the total momentum before an interaction equals the total after. For a perfectly inelastic collision the objects stick together: m₁v₁ + m₂v₂ = (m₁ + m₂)v′.

Worked examples

Avoid these mistakes

• Momentum conservation does not automatically imply kinetic-energy conservation.

Words to know

momentum

The 'quantity of motion' of an object, p = mv; a vector measured in kilogram-metres per second (kg·m/s).

impulse

The product of force and the time it acts, J = FΔt; it equals the change in momentum, Δp.

conservation of momentum

When no net external force acts, the total momentum of a system is the same before and after an interaction.

elastic collision

A collision in which both momentum and kinetic energy are conserved.

inelastic collision

A collision in which momentum is conserved but some kinetic energy is lost; in a perfectly inelastic collision the objects stick together.

Descartes, Huygens, and the birth of conservation of momentum

Long before the word 'energy' was well understood, René Descartes proposed in the 1600 s that the total 'quantity of motion' in the universe is fixed. His early version was flawed because he ignored direction, but the Dutch physicist Christiaan Huygens corrected it by treating motion as a directed (vector) quantity and worked out the rules for elastic collisions. When Newton later published his laws, conservation of momentum fell out naturally from his third law. It remains one of the most robust principles in all of physics, still holding in the quantum and relativistic realms where much of Newton's mechanics needs revision.

Misconception: momentum and kinetic energy are not the same 'amount of motion'

Both p = mv and KE = ½mv² measure something about motion, so students often blur them — but they behave very differently. Momentum is a vector (direction matters, and opposite motions can cancel) and it is conserved in every collision. Kinetic energy is a scalar (always positive) and is conserved only in elastic collisions. In a head-on crash where two identical cars moving at equal speeds stick together and stop, the total momentum was zero before and after (conserved), while all the kinetic energy vanished into crumpling metal and heat. Keeping the two quantities separate is the key to analyzing any collision.

Impulse and the physics of staying alive in a crash

Impulse, J = FΔt = Δp, is the single most important idea in crash safety. Your change in momentum in an accident is fixed the instant the crash begins — but the force you feel depends on how long the stop takes. Crumple zones let the front of a car collapse over a longer distance and time; airbags cushion your head over tens of milliseconds instead of one; even a boxer 'rolling with' a punch or a gymnast bending their knees on landing is stretching Δt to shrink F = Δp/Δt. The same equation explains why landing on concrete hurts far more than landing on a crash mat, even from the same height.

Rockets: pushing on nothing at all

A common myth is that rockets fly by 'pushing against the air.' They don't — they work best in the vacuum of space, and the reason is pure conservation of momentum. A rocket hurls a stream of exhaust gas backward at enormous speed; to keep the system's total momentum constant, the rocket must gain an equal and opposite momentum forward. No ground, no air, and nothing external to push against is required — only the exhaust being thrown one way and the rocket recoiling the other. The same principle lets an astronaut drift back to a spacecraft by throwing a tool in the opposite direction.

In the real world

Rockets work by conservation of momentum: they hurl exhaust gas backward at high speed, and the equal-and-opposite forward momentum drives the rocket ahead — which is why a rocket can accelerate in the vacuum of space with nothing to push against.

Exit check

2 kg at 3 m/s sticks to 1 kg atrest. Find final velocity.

Impulse & Collision Evidence

Learning goals

• I can use a force-time area to calculate signed momentum change.
• I can explain collision time, average force, and system conservation.

Readings for this unit

• A slower cart has not destroyed its energy

Momentum p=mv is a vector quantity. Impulse J=∫Fdt changes momentum; for a known average force, J=FavgΔt=Δp. A force-time graph’s signed area gives impulse. It is the change in momentum rather than change in speed alone that matters when direction reverses.

Choose signs before subtracting: Δp=m(vfinal−vinitial). A ball moving right and bouncing left has a larger momentum change magnitude than an otherwise similar ball merely stopping. A negative result identifies the impulse direction under the selected convention.

A longer stopping time can reduce average force for the same momentum change. Padding or a crumple zone increases interaction time and distance; it does not remove the need for momentum change. Peak force can differ greatly from average force, so an average-force calculation does not describe every instant.

Momentum conservation applies to a system when net external impulse is negligible during the interval. Internal forces cancel between interacting bodies by Newton’s third law. Individual object momentum changes while total system momentum can stay the same. Define the system before claiming conservation.

Kinetic energy need not be conserved in an inelastic collision; some energy becomes deformation, vibration, or thermal energy. Total energy remains accounted for in an appropriately inclusive system. In an elastic ideal collision, both total momentum and total kinetic energy are conserved.

Use supplied force-time data or gentle tabletop models. Collision experiments with people, fast projectiles, or fragile objects are unnecessary. Evaluate whether friction or an external support had enough impulse to undermine the isolated-system approximation.

Worked examples

Avoid these mistakes

• A bounce requires direction signs; do not subtract speed magnitudes only.

Words to know

impulse

Time integral of force, equal to momentum change.

external impulse

Impulse from forces exerted by objects outside the chosen system.

In the real world

Use the chapter investigation to connect this unit to measured evidence and a practical decision.

Exit check

Why distinguish average from peak collision force?