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This free lesson on MarkScheme covers syllabus point A.3 (Work, energy and power) for IB Diploma Physics (Higher Level). It includes visual revision, key points, and links to criterion-based practice marking at markscheme.app/ib/courses/physics-hl/a-3-work-energy-and-power.

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Work, energy and power — IB Physics HL

Work, energy and power

Almost every mechanics problem can be solved twice: once with forces and acceleration, and once — often far more quickly — with energy. Energy is the universal accounting currency of physics: it flows between kinetic, gravitational, elastic and thermal stores, but the total in an isolated system never changes. This lesson builds that toolkit from the ground up: how a force transfers energy through work, the three energy stores you meet in this course, the principle of conservation of energy, and finally power and efficiency, which tell you how fast and how effectively energy is transferred. Because A.3 is calculation-heavy, every worked example is laid out exactly the way marks are awarded — because in this topic, HOW you show your working decides how many marks you keep.

BY THE END, YOU CAN…

exam checklist — tick these off

1Calculate the work done by a constant force using $W = Fs\cos\theta$ , and find work as the area under a force–displacement graph.
2Calculate kinetic energy $E_k = \tfrac{1}{2}mv^2$ , change in gravitational potential energy $\Delta E_p = mg\Delta h$ , and elastic potential energy $E_e = \tfrac{1}{2}kx^2$ .
3Apply the principle of conservation of energy to track energy transfers, including problems with and without friction.
4Calculate power as $P = W/t = Fv$ and efficiency as useful output divided by total input.
5Present working so that method marks and error-carried-forward (ECF) protect the final answer.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Energy is a Currency You Cannot Destroy

Energy is like money in a sealed system: it moves from one account to another — motion, height, stretch, heat — but the total never changes. Work is a transfer of energy by a force, and power is how fast that transfer happens.

THINK OF IT LIKE…

Picture a skateboarder at the top of a ramp. At the top they have a full 'height account' (gravitational potential energy) and an empty 'speed account' (kinetic energy). Rolling down, money is transferred from the height account into the speed account — but no money is created or lost. If the ramp is rough, some money is quietly siphoned off into a 'heat account' by friction, so the speed account fills a little less. Add up every account at the bottom and you get exactly what you started with at the top.

1
Choose the system and identify every energy store involved — kinetic, gravitational, elastic, and any thermal energy lost to friction.
2
Write the total energy at the start and the total energy at the end; conservation of energy says they are equal.
3
For a single force, calculate the energy it transfers using $W = Fs\cos\theta$ , or the area under a force–displacement graph.
4
If the question asks how fast the transfer happens, use power $P = W/t = Fv$ ; if it asks how much is usefully transferred, use efficiency = useful ÷ total.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

Figure PhET simulation1 / 4

Loading simulation…

Fetching from PhET…

Energy Skate Park: Basics

Show the bar graph and release the skater: kinetic and potential energy trade places while the total stays level. Add friction and thermal energy takes the rest.

If the embed does not load, use Open on PhET.

Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)

Step 1

Choose the system and identify every energy store involved — kinetic, gravitational, elastic, and any thermal energy lost to friction.

28 more simulations for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

28 simulations · 5 to start with

Start herein this order — each one shows a different piece of the topic

PhETStart here · 19702 6.2 · IB A.3
Hooke’s Law
Stretch a spring and watch the area under the force–extension graph become stored energy.
Why this one: Stretch the spring and watch the triangle under the F-x graph become the stored half-k-x-squared.
Try this
1. Go to “Energy”; set k = 200 N/m and pull to 0.3 m — read the energy and check ½kx².
2. Compare the force–extension plot with the energy bar as you drag slowly.
3. Put two springs in series, then parallel — which combination stores more at the same force?
Look for Elastic potential energy = area under F–x = ½Fx = ½kx².
Open on PhET
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
The Physics ClassroomStart here · 29702 5.1 · 9702 5.2 · IB A.3
Kinetic Energy
Relate the work done on an object to the kinetic energy it acquires, and see what friction changes
Why this one: Compare the work the force does with the kinetic energy gained, then add friction and find the gap.
Try this
1. Apply a force and read the kinetic energy gained.
2. Add friction and compare the kinetic energy.
Look for Net work done equals the change in kinetic energy.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomStart here · 39702 5.1 · 9702 5.2 · IB A.3
It's All Uphill
Drive a car up a gently-sloped hill and a steep hill to the same summit and compare the work and energy
Why this one: Both hills give the same mg-delta-h; the gentle one needs less force over more distance.
Try this
1. Drive up the gentle hill and read the work done.
2. Drive up the steep hill to the same summit.
3. Compare the two values.
Look for Work done against gravity depends only on the height gained, not on the slope.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomStart here · 49702 5.1 · 9702 5.2 · IB A.3
Stopping Distance
Skid a car to a stop from different speeds and compare the stopping distances
Why this one: Double the speed and the skid quadruples because friction must remove four times the KE.
Try this
1. Stop from one speed and read the distance.
2. Double the speed and stop again.
3. Compare the two distances.
Look for Stopping distance scales with the square of the speed.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
3JCN PhysicsStart here · 59702 5.1 · 9702 5.2 · IB A.3
CoE on an Inclined Ramp
Slide a block down a ramp with chosen height and friction; track energy transfers
Why this one: Slide the block with friction and track where the lost mechanical energy goes.
Try this
1. Set a height with no friction and slide the block; track the energy transfers.
2. Add friction and compare the KE at the bottom.
Look for Without friction PE converts fully to KE; with friction the missing KE equals the work done against friction.
Open on 3JCN Physics
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0

More simulations23 more on this topic — core ones first

PhETCore9702 5.1 · IB A.3
Energy Skate Park
Build a track, drop a skater and watch kinetic, potential and thermal energy trade places.
Try this
1. Start the skater at 6 m with friction off; show the bar graph — where is KE largest?
2. Add friction and watch the thermal bar grow while the total stays flat.
3. Change the skater’s mass — does the top speed at the bottom change?
Look for Total energy is constant; friction converts mechanical energy to thermal, never destroys it.
Open on PhET
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
PhETCoreJava · best on a laptop9702 5.1 · 3.1 · 4.2 · IB A.3 · A.2
The Ramp
Push objects up a ramp; vary the angle, friction and mass and read the work done, energy bar charts and force graphs.
Try this
1. Push a crate to the top at 15° with friction off — read the work done and the GPE gained.
2. Turn friction on and repeat — the extra work shows up as thermal energy.
3. Steepen the ramp — more force per metre, but the same GPE at the top.
Look for Work done = gain in GPE + energy lost to friction; the ramp trades force for distance.
Open on PhET
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
oPhysicsCore9702 3.3 · 9702 5.1 · IB A.2
Momentum & Energy: Elastic and Inelastic Collisions
1D collision of two masses: set masses, velocities and elasticity; compare momentum and KE before/after
Try this
1. Set equal masses, one at rest, fully elastic; compare the velocities after.
2. Make the collision fully inelastic with the same setup.
3. Give one mass twice the other and run both elasticities.
Look for Momentum is the same before and after in every case; kinetic energy is conserved only when the collision is elastic.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 3.3 · 9702 5.1 · IB A.2
The Ballistic Pendulum
Ballistic pendulum: set bullet/block masses and bullet speed; see inelastic collision then rise height
Try this
1. Set a bullet speed and read the rise height.
2. Double the bullet speed and compare the rise height.
3. Increase the block mass with the same bullet speed.
Look for Momentum is conserved in the collision and energy afterwards, so the rise height grows with the square of the bullet speed.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 3.3 · 9702 5.1 · IB A.2
Ballistic Pendulum "Quiz"
Ballistic pendulum quiz: given masses and rise height, compute the bullet speed and check it
Try this
1. Read the masses and rise height and compute the speed just after impact.
2. Use momentum conservation to find the bullet speed and check it.
3. Reset for a new set of values and repeat.
Look for The speed just after impact comes from the rise height, and the bullet speed from momentum conservation.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 5.1 · 9702 5.2 · 9702 17.2
Conservation of Mechanical Energy: Mass on a Vertical Spring
Mass oscillating on a vertical spring with live KE / GPE / EPE bar graphs; adjust mass and spring constant
Try this
1. Watch the three bars through one full oscillation.
2. Increase the mass and compare the bar heights.
3. Increase the spring constant and compare the period.
Look for KE, GPE and EPE trade against each other while their total stays constant.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)

Key formulas

Tap any symbol to reveal exactly what it means and its units.

KEY RELATION

$W = Fs\cos\theta$

KEY RELATION

$E_k = \tfrac{1}{2}mv^2 \qquad \Delta E_p = mg\Delta h \qquad E_e = \tfrac{1}{2}kx^2$

Tap a symbol — great for exam definitions

KEY RELATION

$mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$

KEY RELATION

$P = \frac{W}{t} = \frac{\Delta E}{t} = Fv$

KEY RELATION

$\eta = \frac{\text{useful energy output}}{\text{total energy input}} = \frac{\text{useful power output}}{\text{total power input}}$

Full topic notes

Formal explanation with the rigour you need for the exam.

Work: transferring energy with a force

In physics, work has a precise meaning: work is done when a force causes a displacement, and it is the mechanism by which energy is transferred to or from an object. Crucially, only the component of the force acting ALONG the displacement does work. A force applied to an object that does not move does no work; a force perpendicular to the motion — such as the normal force on a book sliding across a table — does no work either.

$W = Fs\cos\theta$

The formula $W = Fs\cos\theta$ only applies to a constant force. When the force VARIES with displacement — as a spring's force does — the work done is instead the AREA under the force–displacement ( $F$ – $s$ ) graph. This graphical view is completely general: for a constant force the area is simply a rectangle of height $F$ and width $s$ , recovering $W = Fs$ , while for a spring obeying $F = kx$ the area is a triangle, which is where the elastic energy formula comes from.

Work ( $W$ ) is a scalar quantity measured in joules (J); 1 J = 1 N m.
$F$ is the magnitude of the force (N) and $s$ is the magnitude of the displacement (m).
$\theta$ is the angle between the force and the displacement, so $F\cos\theta$ is the component of force along the motion.
Work is zero when $s = 0$ , when $F = 0$ , or when $\theta = 90^\circ$ ; work is negative when $\theta > 90^\circ$ (the force opposes the motion, e.g. friction).

EXAM TIP

If a question shows an $F$ – $s$ graph, do not reach for $W = Fs$ . Find the AREA under the line instead — split it into rectangles and triangles if the shape is awkward. Marks are routinely lost by students who read a single force value off the axis and multiply, ignoring that the force changes.

Not clear?

The three energy stores

For mechanics you need three stores of energy. Kinetic energy ( $E_k$ ) is the energy of motion — it grows with the square of the speed, so a car at 40 m s⁻¹ carries four times the kinetic energy it has at 20 m s⁻¹. Gravitational potential energy ( $E_p$ ) is the energy of position in a gravitational field; lifting a mass raises it, and only the change in VERTICAL height matters. Elastic potential energy ( $E_e$ ) is the energy stored in a stretched or compressed spring — it is the area under the spring's force–extension graph.

$E_k = \tfrac{1}{2}mv^2 \qquad \Delta E_p = mg\Delta h \qquad E_e = \tfrac{1}{2}kx^2$

$E_k = \tfrac{1}{2}mv^2$ : kinetic energy depends on the SQUARE of speed — doubling $v$ quadruples $E_k$ .
$\Delta E_p = mg\Delta h$ : use the vertical height change $\Delta h$ , not the distance along a ramp; $g = 9.81$ m s⁻² near Earth's surface.
$E_e = \tfrac{1}{2}kx^2$ : $k$ is the spring constant (N m⁻¹) and $x$ is the extension or compression (m); this is the triangular area under $F = kx$ .
All three are measured in joules (J).

Not clear?

Conservation of energy and energy transfers

The principle of conservation of energy states that in an isolated system the total energy is constant — energy is transferred between stores but is never created or destroyed. This is the single most powerful idea in mechanics, because it lets you jump directly from a starting state to a finishing state without tracking the forces at every instant. For a mass falling or sliding down a frictionless slope, gravitational potential energy is transferred entirely to kinetic energy: $mg\Delta h = \tfrac{1}{2}mv^2$ . Notice the mass cancels, so the speed at the bottom does not depend on how heavy the object is.

When friction or air resistance is present, energy is not lost — it is transferred to thermal energy in the surfaces and surroundings. The bookkeeping simply gains one more term: the energy at the start equals the useful kinetic energy at the end PLUS the energy transferred to thermal energy. This is why a real object always arrives at the bottom of a rough slope slower than the frictionless prediction: some of its potential energy has been siphoned off as heat.

$mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$

EXAM TIP

The energy method avoids acceleration entirely. If you find yourself reaching for $v^2 = u^2 + 2as$ on a slope and the angle or path is awkward, switch to energy: equate the store you start with to the stores you finish with. Just remember to include a thermal-energy term whenever friction or drag is mentioned.

Not clear?

Power: the rate of energy transfer

Two machines can transfer the same energy, but the one that does it faster is more powerful. Power is the rate of doing work, or equivalently the rate of transferring energy, measured in watts (W), where one watt is one joule per second. For an object moving at a steady velocity $v$ under a constant force $F$ , a compact form is available: since $W = Fs$ and $v = s/t$ , we get $P = W/t = Fs/t = Fv$ . This is ideal for steady-state motion such as a vehicle cruising against drag.

$P = \frac{W}{t} = \frac{\Delta E}{t} = Fv$

Not clear?

Efficiency: how much of the input is useful

No real machine transfers all of its input energy into the store you want — some always ends up as unwanted thermal energy or sound. Efficiency measures how much of the input is usefully transferred: it is the ratio of useful output to total input, and because some energy is always wasted, it is always less than 1, or below 100%. You can compute it from energies or from powers, since power is just energy per unit time — as long as you compare like with like.

$\eta = \frac{\text{useful energy output}}{\text{total energy input}} = \frac{\text{useful power output}}{\text{total power input}}$

Efficiency is a dimensionless ratio, often expressed as a percentage (multiply by 100).
It is always less than 100% for a real machine — a value of 100% or more signals an error.
Use either energies (J) or powers (W), but never mix one on top and the other below.
The 'wasted' energy is not destroyed — it is transferred to the surroundings, usually as thermal energy.

Not clear?

Common mistakes examiners penalise

Forgetting $\cos\theta$ in work — work needs the displacement in the DIRECTION of the force. If the force is at an angle, you must use $W = Fs\cos\theta$ , not $W = Fs$ . Holding a weight still, or a force perpendicular to motion, does ZERO work.
Saying energy is 'used up' or 'lost' — energy is conserved, never destroyed. When friction slows an object, state that the kinetic energy is TRANSFERRED to thermal energy in the surfaces and surroundings.
Using distance along a slope for $\Delta h$ — gravitational potential energy uses the VERTICAL height change, not the length of the ramp. Take the vertical component.
Forgetting the friction term in conservation of energy — on a rough slope, $mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$ . Ignoring the thermal term overestimates the final speed.
Confusing power with energy — power is energy per second (W), energy is the total transferred (J). Doing the same job faster means more power, not more work.
Quoting an efficiency of 100% or more — a real machine always wastes some energy, so efficiency is always below 100%. A value at or above 100% means useful and total inputs have been swapped or a wasted term has been missed.
Forgetting kinetic energy scales with $v^2$ — doubling the speed quadruples $E_k$ ; halving the speed quarters it. Do not treat $E_k$ as if it were proportional to $v$ .

Not clear?

Model answer — marked the way our engine marks it

This is the showcase for a calculation topic. In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on does not cost you every mark that follows, PROVIDED your method is written down. Notice how the marks here split between the numerical calculation and the physical REASONING about air resistance: the reasoning marks are earned by explanation, not arithmetic. Study how each mark is earned by a specific line.

Not clear?

Where this leads

Conservation of energy is not confined to this topic — it is the spine of the whole course. Momentum and collisions revisit kinetic energy to distinguish elastic from inelastic; simple harmonic motion swaps energy continually between elastic and kinetic stores; thermal physics tracks the very thermal energy that friction produces here; and in the study of fields, potential energy generalises from $mg\Delta h$ to gravitational and electric potentials. Master the habit of accounting — write every store at the start, every store at the end, set them equal — and problem after problem across the syllabus becomes a variation on a method you already own.

Not clear?

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

reveal slowly — mark each step

EXAMPLE 1

A person pulls a 20.0 kg suitcase 15.0 m across a flat floor using a strap at 40° above the horizontal, with a constant force of 50.0 N. (a) Calculate the work done by the person on the suitcase. (b) The suitcase starts from rest and friction is negligible. Using the work–energy principle, find its final speed. [5]

1
(a) Work done by the pull. Only the horizontal component of the force acts along the motion, so use $W = Fs\cos\theta$ . $W = (50.0)(15.0)\cos 40^\circ = 750 \times 0.766 = 575$ J. [M1 method with $\cos\theta$ , A1 answer to 3 s.f.]

EXAMPLE 2

A 2.0 kg box is released from rest at the top of a ramp and slides down a vertical drop of 3.0 m. (a) Assuming the ramp is frictionless, calculate the box's speed at the bottom. (b) In reality the box reaches the bottom at 6.0 m s⁻¹. Calculate the energy transferred to thermal energy by friction. (Use $g = 9.81$ m s⁻².) [5]

1
(a) Frictionless case — energy conservation. All the gravitational potential energy becomes kinetic energy: $mg\Delta h = \tfrac{1}{2}mv^2$ . [M1] The mass cancels: $v = \sqrt{2g\Delta h} = \sqrt{2 \times 9.81 \times 3.0} = \sqrt{58.9} = 7.67$ m s⁻¹. [A1]

EXAMPLE 3

An electric winch lifts a 65 kg crate vertically at a constant speed of 0.40 m s⁻¹. The winch draws 350 W of electrical power. (a) Calculate the useful mechanical power output. (b) Calculate the efficiency of the winch. (Use $g = 9.81$ m s⁻².) [4]

1
(a) Useful output power. At constant speed the winch's lifting force balances the weight, $F = mg = 65 \times 9.81 = 637.65$ N. Useful power lifts the crate: $P = Fv = 637.65 \times 0.40 = 255$ W. [M1 using $P = Fv$ (or $mgv$), A1]

EXAMPLE 4

A 0.50 kg ball is dropped from rest at a height of 8.0 m. (a) Using energy conservation, calculate its speed just before it hits the ground. (b) Explain how your answer would change if air resistance were significant. (Use $g = 9.81$ m s⁻².) [4]

1
Model answer — full working.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAWork, energy and power

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Key terms for this topic — skim now; the Check step will test them.

Work done by a constant force: $W = Fs\cos\theta$ , where $\theta$ is the angle between the force and the displacement. Only the component of force ALONG the displacement does work. Work is a scalar, measured in joules (J).
Work as area under a graph: For a variable force, the work done equals the AREA under the force–displacement ( $F$ – $s$ ) graph. This is how the energy stored in a spring, $\tfrac{1}{2}kx^2$ , is obtained — the area of the triangle under $F = kx$ .
Kinetic energy: $E_k = \tfrac{1}{2}mv^2$ — the energy of a moving mass. Note it depends on the SQUARE of speed, so doubling the speed quadruples the kinetic energy.
Change in gravitational potential energy: $\Delta E_p = mg\Delta h$ , where $\Delta h$ is the change in VERTICAL height, not distance along a slope. Units: joules (J).
Elastic potential energy: $E_e = \tfrac{1}{2}kx^2$ — the energy stored in a spring (obeying Hooke's law $F = kx$ ) that is stretched or compressed by an extension $x$ . It is the area under the force–extension graph.
Principle of conservation of energy: In an isolated system the total energy is constant: energy is transferred between stores but never created or destroyed. Total energy before = total energy after.
Power: $P = \dfrac{W}{t} = \dfrac{\Delta E}{t}$ — the RATE of doing work or transferring energy, measured in watts (W), where 1 W = 1 J s⁻¹.
Power at constant velocity: $P = Fv$ , derived from $P = W/t = Fs/t = F(s/t)$ . Useful when a force $F$ moves an object at speed $v$ , e.g. a car overcoming drag at steady speed.
Efficiency: $\eta = \dfrac{\text{useful energy (or power) out}}{\text{total energy (or power) in}}$ . A dimensionless ratio, often written as a percentage. It is ALWAYS less than 1 (100%) for a real machine.
Negative work: Work is negative when a force (or its component) opposes the displacement, e.g. friction. Negative work REMOVES kinetic energy from the object; it does not vanish, it becomes thermal energy.

09Remember

Name it

Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.

1 of 6 · set 1 of 2

Which of these is being described?

$E_e = \tfrac{1}{2}kx^2$ — the energy stored in a spring (obeying Hooke's law $F = kx$ ) that is stretched or compressed by an extension $x$ . It is the area under the force–extension graph.

Choose the term this describes.

10Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 5 answered

11Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

12Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

close the tab only after these stick

Work ( $W$ ) is a scalar quantity measured in joules (J); 1 J = 1 N m.
$F$ is the magnitude of the force (N) and $s$ is the magnitude of the displacement (m).
$\theta$ is the angle between the force and the displacement, so $F\cos\theta$ is the component of force along the motion.
Work is zero when $s = 0$ , when $F = 0$ , or when $\theta = 90^\circ$ ; work is negative when $\theta > 90^\circ$ (the force opposes the motion, e.g. friction).

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Get a Paper 2 energy calculation marked: use conservation of energy to find a speed, then reason about air resistance

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

IB Physics Data Booklet↗

Frequently asked

Can work be negative, and what does that mean physically?

What is the difference between energy and power?

Why does the conservation-of-energy method avoid needing the acceleration or the forces at every instant?

Why can efficiency never reach 100% for a real machine?

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Get a Paper 2 energy calculation marked: use conservation of energy to find a speed, then reason about air resistance on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUSA.2 Forces and momentum NEXT →A.4 Rigid body mechanics

⚛

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Work, energy and power

BY THE END, YOU CAN…

exam checklist — tick these off

1Calculate the work done by a constant force using $W = Fs\cos\theta$ , and find work as the area under a force–displacement graph.
2Calculate kinetic energy $E_k = \tfrac{1}{2}mv^2$ , change in gravitational potential energy $\Delta E_p = mg\Delta h$ , and elastic potential energy $E_e = \tfrac{1}{2}kx^2$ .
3Apply the principle of conservation of energy to track energy transfers, including problems with and without friction.
4Calculate power as $P = W/t = Fv$ and efficiency as useful output divided by total input.
5Present working so that method marks and error-carried-forward (ECF) protect the final answer.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Energy is a Currency You Cannot Destroy

THINK OF IT LIKE…

1
Choose the system and identify every energy store involved — kinetic, gravitational, elastic, and any thermal energy lost to friction.
2
Write the total energy at the start and the total energy at the end; conservation of energy says they are equal.
3
For a single force, calculate the energy it transfers using $W = Fs\cos\theta$ , or the area under a force–displacement graph.
4
If the question asks how fast the transfer happens, use power $P = W/t = Fv$ ; if it asks how much is usefully transferred, use efficiency = useful ÷ total.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

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Energy Skate Park: Basics

Show the bar graph and release the skater: kinetic and potential energy trade places while the total stays level. Add friction and thermal energy takes the rest.

If the embed does not load, use Open on PhET.

Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)

Step 1

Choose the system and identify every energy store involved — kinetic, gravitational, elastic, and any thermal energy lost to friction.

28 more simulations for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

28 simulations · 5 to start with

Start herein this order — each one shows a different piece of the topic

PhETStart here · 19702 6.2 · IB A.3
Hooke’s Law
Stretch a spring and watch the area under the force–extension graph become stored energy.
Why this one: Stretch the spring and watch the triangle under the F-x graph become the stored half-k-x-squared.
Try this
1. Go to “Energy”; set k = 200 N/m and pull to 0.3 m — read the energy and check ½kx².
2. Compare the force–extension plot with the energy bar as you drag slowly.
3. Put two springs in series, then parallel — which combination stores more at the same force?
Look for Elastic potential energy = area under F–x = ½Fx = ½kx².
Open on PhET
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
The Physics ClassroomStart here · 29702 5.1 · 9702 5.2 · IB A.3
Kinetic Energy
Relate the work done on an object to the kinetic energy it acquires, and see what friction changes
Why this one: Compare the work the force does with the kinetic energy gained, then add friction and find the gap.
Try this
1. Apply a force and read the kinetic energy gained.
2. Add friction and compare the kinetic energy.
Look for Net work done equals the change in kinetic energy.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomStart here · 39702 5.1 · 9702 5.2 · IB A.3
It's All Uphill
Drive a car up a gently-sloped hill and a steep hill to the same summit and compare the work and energy
Why this one: Both hills give the same mg-delta-h; the gentle one needs less force over more distance.
Try this
1. Drive up the gentle hill and read the work done.
2. Drive up the steep hill to the same summit.
3. Compare the two values.
Look for Work done against gravity depends only on the height gained, not on the slope.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomStart here · 49702 5.1 · 9702 5.2 · IB A.3
Stopping Distance
Skid a car to a stop from different speeds and compare the stopping distances
Why this one: Double the speed and the skid quadruples because friction must remove four times the KE.
Try this
1. Stop from one speed and read the distance.
2. Double the speed and stop again.
3. Compare the two distances.
Look for Stopping distance scales with the square of the speed.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
3JCN PhysicsStart here · 59702 5.1 · 9702 5.2 · IB A.3
CoE on an Inclined Ramp
Slide a block down a ramp with chosen height and friction; track energy transfers
Why this one: Slide the block with friction and track where the lost mechanical energy goes.
Try this
1. Set a height with no friction and slide the block; track the energy transfers.
2. Add friction and compare the KE at the bottom.
Look for Without friction PE converts fully to KE; with friction the missing KE equals the work done against friction.
Open on 3JCN Physics
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0

More simulations23 more on this topic — core ones first

PhETCore9702 5.1 · IB A.3
Energy Skate Park
Build a track, drop a skater and watch kinetic, potential and thermal energy trade places.
Try this
1. Start the skater at 6 m with friction off; show the bar graph — where is KE largest?
2. Add friction and watch the thermal bar grow while the total stays flat.
3. Change the skater’s mass — does the top speed at the bottom change?
Look for Total energy is constant; friction converts mechanical energy to thermal, never destroys it.
Open on PhET
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
PhETCoreJava · best on a laptop9702 5.1 · 3.1 · 4.2 · IB A.3 · A.2
The Ramp
Push objects up a ramp; vary the angle, friction and mass and read the work done, energy bar charts and force graphs.
Try this
1. Push a crate to the top at 15° with friction off — read the work done and the GPE gained.
2. Turn friction on and repeat — the extra work shows up as thermal energy.
3. Steepen the ramp — more force per metre, but the same GPE at the top.
Look for Work done = gain in GPE + energy lost to friction; the ramp trades force for distance.
Open on PhET
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
oPhysicsCore9702 3.3 · 9702 5.1 · IB A.2
Momentum & Energy: Elastic and Inelastic Collisions
1D collision of two masses: set masses, velocities and elasticity; compare momentum and KE before/after
Try this
1. Set equal masses, one at rest, fully elastic; compare the velocities after.
2. Make the collision fully inelastic with the same setup.
3. Give one mass twice the other and run both elasticities.
Look for Momentum is the same before and after in every case; kinetic energy is conserved only when the collision is elastic.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 3.3 · 9702 5.1 · IB A.2
The Ballistic Pendulum
Ballistic pendulum: set bullet/block masses and bullet speed; see inelastic collision then rise height
Try this
1. Set a bullet speed and read the rise height.
2. Double the bullet speed and compare the rise height.
3. Increase the block mass with the same bullet speed.
Look for Momentum is conserved in the collision and energy afterwards, so the rise height grows with the square of the bullet speed.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 3.3 · 9702 5.1 · IB A.2
Ballistic Pendulum "Quiz"
Ballistic pendulum quiz: given masses and rise height, compute the bullet speed and check it
Try this
1. Read the masses and rise height and compute the speed just after impact.
2. Use momentum conservation to find the bullet speed and check it.
3. Reset for a new set of values and repeat.
Look for The speed just after impact comes from the rise height, and the bullet speed from momentum conservation.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 5.1 · 9702 5.2 · 9702 17.2
Conservation of Mechanical Energy: Mass on a Vertical Spring
Mass oscillating on a vertical spring with live KE / GPE / EPE bar graphs; adjust mass and spring constant
Try this
1. Watch the three bars through one full oscillation.
2. Increase the mass and compare the bar heights.
3. Increase the spring constant and compare the period.
Look for KE, GPE and EPE trade against each other while their total stays constant.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)

Key formulas

Tap any symbol to reveal exactly what it means and its units.

KEY RELATION

$W = Fs\cos\theta$

KEY RELATION

$E_k = \tfrac{1}{2}mv^2 \qquad \Delta E_p = mg\Delta h \qquad E_e = \tfrac{1}{2}kx^2$

Tap a symbol — great for exam definitions

KEY RELATION

$mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$

KEY RELATION

$P = \frac{W}{t} = \frac{\Delta E}{t} = Fv$

KEY RELATION

$\eta = \frac{\text{useful energy output}}{\text{total energy input}} = \frac{\text{useful power output}}{\text{total power input}}$

Full topic notes

Formal explanation with the rigour you need for the exam.

Work: transferring energy with a force

$W = Fs\cos\theta$

Work ( $W$ ) is a scalar quantity measured in joules (J); 1 J = 1 N m.
$F$ is the magnitude of the force (N) and $s$ is the magnitude of the displacement (m).
$\theta$ is the angle between the force and the displacement, so $F\cos\theta$ is the component of force along the motion.
Work is zero when $s = 0$ , when $F = 0$ , or when $\theta = 90^\circ$ ; work is negative when $\theta > 90^\circ$ (the force opposes the motion, e.g. friction).

EXAM TIP

Not clear?

The three energy stores

$E_k = \tfrac{1}{2}mv^2 \qquad \Delta E_p = mg\Delta h \qquad E_e = \tfrac{1}{2}kx^2$

$E_k = \tfrac{1}{2}mv^2$ : kinetic energy depends on the SQUARE of speed — doubling $v$ quadruples $E_k$ .
$\Delta E_p = mg\Delta h$ : use the vertical height change $\Delta h$ , not the distance along a ramp; $g = 9.81$ m s⁻² near Earth's surface.
$E_e = \tfrac{1}{2}kx^2$ : $k$ is the spring constant (N m⁻¹) and $x$ is the extension or compression (m); this is the triangular area under $F = kx$ .
All three are measured in joules (J).

Not clear?

Conservation of energy and energy transfers

$mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$

EXAM TIP

Not clear?

Power: the rate of energy transfer

$P = \frac{W}{t} = \frac{\Delta E}{t} = Fv$

Not clear?

Efficiency: how much of the input is useful

$\eta = \frac{\text{useful energy output}}{\text{total energy input}} = \frac{\text{useful power output}}{\text{total power input}}$

Efficiency is a dimensionless ratio, often expressed as a percentage (multiply by 100).
It is always less than 100% for a real machine — a value of 100% or more signals an error.
Use either energies (J) or powers (W), but never mix one on top and the other below.
The 'wasted' energy is not destroyed — it is transferred to the surroundings, usually as thermal energy.

Not clear?

Common mistakes examiners penalise

Forgetting $\cos\theta$ in work — work needs the displacement in the DIRECTION of the force. If the force is at an angle, you must use $W = Fs\cos\theta$ , not $W = Fs$ . Holding a weight still, or a force perpendicular to motion, does ZERO work.
Saying energy is 'used up' or 'lost' — energy is conserved, never destroyed. When friction slows an object, state that the kinetic energy is TRANSFERRED to thermal energy in the surfaces and surroundings.
Using distance along a slope for $\Delta h$ — gravitational potential energy uses the VERTICAL height change, not the length of the ramp. Take the vertical component.
Forgetting the friction term in conservation of energy — on a rough slope, $mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$ . Ignoring the thermal term overestimates the final speed.
Confusing power with energy — power is energy per second (W), energy is the total transferred (J). Doing the same job faster means more power, not more work.
Quoting an efficiency of 100% or more — a real machine always wastes some energy, so efficiency is always below 100%. A value at or above 100% means useful and total inputs have been swapped or a wasted term has been missed.
Forgetting kinetic energy scales with $v^2$ — doubling the speed quadruples $E_k$ ; halving the speed quarters it. Do not treat $E_k$ as if it were proportional to $v$ .

Not clear?

Model answer — marked the way our engine marks it

Not clear?

Where this leads

Not clear?

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

reveal slowly — mark each step

EXAMPLE 1

1
(a) Work done by the pull. Only the horizontal component of the force acts along the motion, so use $W = Fs\cos\theta$ . $W = (50.0)(15.0)\cos 40^\circ = 750 \times 0.766 = 575$ J. [M1 method with $\cos\theta$ , A1 answer to 3 s.f.]

EXAMPLE 2

1
(a) Frictionless case — energy conservation. All the gravitational potential energy becomes kinetic energy: $mg\Delta h = \tfrac{1}{2}mv^2$ . [M1] The mass cancels: $v = \sqrt{2g\Delta h} = \sqrt{2 \times 9.81 \times 3.0} = \sqrt{58.9} = 7.67$ m s⁻¹. [A1]

EXAMPLE 3

1
(a) Useful output power. At constant speed the winch's lifting force balances the weight, $F = mg = 65 \times 9.81 = 637.65$ N. Useful power lifts the crate: $P = Fv = 637.65 \times 0.40 = 255$ W. [M1 using $P = Fv$ (or $mgv$), A1]

EXAMPLE 4

1
Model answer — full working.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAWork, energy and power

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Key terms for this topic — skim now; the Check step will test them.

Work done by a constant force: $W = Fs\cos\theta$ , where $\theta$ is the angle between the force and the displacement. Only the component of force ALONG the displacement does work. Work is a scalar, measured in joules (J).
Work as area under a graph: For a variable force, the work done equals the AREA under the force–displacement ( $F$ – $s$ ) graph. This is how the energy stored in a spring, $\tfrac{1}{2}kx^2$ , is obtained — the area of the triangle under $F = kx$ .
Kinetic energy: $E_k = \tfrac{1}{2}mv^2$ — the energy of a moving mass. Note it depends on the SQUARE of speed, so doubling the speed quadruples the kinetic energy.
Change in gravitational potential energy: $\Delta E_p = mg\Delta h$ , where $\Delta h$ is the change in VERTICAL height, not distance along a slope. Units: joules (J).
Elastic potential energy: $E_e = \tfrac{1}{2}kx^2$ — the energy stored in a spring (obeying Hooke's law $F = kx$ ) that is stretched or compressed by an extension $x$ . It is the area under the force–extension graph.
Principle of conservation of energy: In an isolated system the total energy is constant: energy is transferred between stores but never created or destroyed. Total energy before = total energy after.
Power: $P = \dfrac{W}{t} = \dfrac{\Delta E}{t}$ — the RATE of doing work or transferring energy, measured in watts (W), where 1 W = 1 J s⁻¹.
Power at constant velocity: $P = Fv$ , derived from $P = W/t = Fs/t = F(s/t)$ . Useful when a force $F$ moves an object at speed $v$ , e.g. a car overcoming drag at steady speed.
Efficiency: $\eta = \dfrac{\text{useful energy (or power) out}}{\text{total energy (or power) in}}$ . A dimensionless ratio, often written as a percentage. It is ALWAYS less than 1 (100%) for a real machine.
Negative work: Work is negative when a force (or its component) opposes the displacement, e.g. friction. Negative work REMOVES kinetic energy from the object; it does not vanish, it becomes thermal energy.

09Remember

Name it

Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.

1 of 6 · set 1 of 2

Which of these is being described?

$E_e = \tfrac{1}{2}kx^2$ — the energy stored in a spring (obeying Hooke's law $F = kx$ ) that is stretched or compressed by an extension $x$ . It is the area under the force–extension graph.

Choose the term this describes.

10Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 5 answered

11Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

12Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

close the tab only after these stick

Work ( $W$ ) is a scalar quantity measured in joules (J); 1 J = 1 N m.
$F$ is the magnitude of the force (N) and $s$ is the magnitude of the displacement (m).
$\theta$ is the angle between the force and the displacement, so $F\cos\theta$ is the component of force along the motion.
Work is zero when $s = 0$ , when $F = 0$ , or when $\theta = 90^\circ$ ; work is negative when $\theta > 90^\circ$ (the force opposes the motion, e.g. friction).

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Get a Paper 2 energy calculation marked: use conservation of energy to find a speed, then reason about air resistance

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

IB Physics Data Booklet↗

Frequently asked

Can work be negative, and what does that mean physically?

What is the difference between energy and power?

Why does the conservation-of-energy method avoid needing the acceleration or the forces at every instant?

Why can efficiency never reach 100% for a real machine?

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUSA.2 Forces and momentum NEXT →A.4 Rigid body mechanics

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In simple terms

Explore the concept

Simulations

Hooke’s Law

Kinetic Energy

It's All Uphill

Stopping Distance

CoE on an Inclined Ramp

Energy Skate Park

The Ramp

Momentum & Energy: Elastic and Inelastic Collisions

The Ballistic Pendulum

Ballistic Pendulum "Quiz"

Conservation of Mechanical Energy: Mass on a Vertical Spring

Key formulas

Full topic notes

Work: transferring energy with a force

The three energy stores

Conservation of energy and energy transfers

Power: the rate of energy transfer

Efficiency: how much of the input is useful

Common mistakes examiners penalise

Model answer — marked the way our engine marks it

Where this leads

Worked examples

How it all connects

Glossary

Name it

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Get a Paper 2 energy calculation marked: use conservation of energy to find a speed, then reason about air resistance

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Work, energy and power

Revise Work, energy and power for IB Physics (A.3)

In simple terms

Explore the concept

Simulations

Hooke’s Law

Kinetic Energy

It's All Uphill

Stopping Distance

CoE on an Inclined Ramp

Energy Skate Park

The Ramp

Momentum & Energy: Elastic and Inelastic Collisions

The Ballistic Pendulum

Ballistic Pendulum "Quiz"

Conservation of Mechanical Energy: Mass on a Vertical Spring

Key formulas

Full topic notes

Work: transferring energy with a force

The three energy stores

Conservation of energy and energy transfers

Power: the rate of energy transfer

Efficiency: how much of the input is useful

Common mistakes examiners penalise

Model answer — marked the way our engine marks it

Where this leads

Worked examples

How it all connects

Glossary

Name it

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Get a Paper 2 energy calculation marked: use conservation of energy to find a speed, then reason about air resistance

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Work, energy and power