Key Idea: This topic is the energy toolkit: forces do work, which is stored as kinetic, gravitational or elastic energy, transferred at a rate set by power, and never destroyed — only shared out and partly wasted as heat. It is examined on both papers: quick definition and 'double the speed' MCQs on Paper 1A, and multi-step calculations (force–distance areas, slide-to-rest, falling-body speeds, spring collisions, power against drag, efficiency) on Paper 2.
📐 The six given equations
Every equation below is given in the data booklet (Theme A.3) — tap any one for the booklet entry. Knowing which to reach for is the skill, so the table after them sorts that out.
- work done = energy transferred (J)
- force applied (N)
- distance moved (m)
- angle between the force and the direction of motion (°)
- kinetic energy — the energy of motion (J)
- mass (kg)
- speed (m s⁻¹)
- momentum, p = mv (kg m s⁻¹)
- change in gravitational PE (J)
- mass (kg)
- gravitational field strength (≈ 9.8 N kg⁻¹ on Earth)
- change in height (m)
- elastic potential energy stored in the spring (J)
- spring constant — the stiffness (N m⁻¹)
- extension or compression from the natural length (m)
- power — energy transferred per second (W = J s⁻¹)
- work done / energy transferred (J)
- time taken (s)
- force (N) — at constant speed, equal to the resistive force
- speed in the direction of the force (m s⁻¹)
- efficiency — the useful fraction (0 to 1; × 100 for a %), no unit
- useful work or power that comes out (J or W)
- total work or power put in (J or W)
🧭 Which one when?
| You're told… | Use | Watch out for |
|---|---|---|
| a force pushing through a distance | W = Fs cos θ | θ is between the force and the motion; force across the motion (90°) does no work |
| a force–distance graph | area under the line = work | rectangle for a flat line, triangle for a sloping spring line — the area is energy (J), not a speed |
| mass and speed (or momentum) | Eₖ = ½mv² (or p²/2m) | square the speed; double v → four times the energy |
| a change of height | ΔEₚ = mgΔh | use the vertical rise only; the mass cancels when you set mgΔh = ½mv² |
| a spring's stiffness and stretch | EH = ½kΔx² | square Δx and convert cm → m first |
| energy (or work) and time | P = ΔW/Δt | answer is in watts (J s⁻¹) |
| a steady speed against a resistive force | P = Fv | F here is the resistive (drag) force, since the speeds aren't changing |
| useful vs total energy | η = useful ÷ total | always ≤ 1; over 100% means you swapped 'useful' and 'total' |
🔁 Conservation, transfers & waste
| Idea | What it says | Typical use |
|---|---|---|
| Work–energy principle | net work done = change in kinetic energy (Wₙₑₜ = ΔEₖ) | a push speeds an object up; friction does negative work that removes Eₖ → slide-to-rest distance |
| Conservation of energy | for a free fall (no air resistance) PE lost = KE gained → mgΔh = ½mv² | landing speeds, pendulum swing speeds — the mass cancels |
| Energy transfer / degradation | energy is never lost, only moved — the wasted branch is almost always thermal energy (heat) | Sankey accounting: useful + wasted branches add up to the input |
When something slides to a stop, friction takes away all its kinetic energy: friction force × distance = Eₖ, so distance = Eₖ ÷ friction force.
✏️ IB-style worked examples
A 3.0 kg trolley starts from rest on a smooth track. A constant net force of 12 N acts on it as it moves 5.0 m, shown on a force–distance graph. (a) State what the area under the graph represents. (b) Find the trolley's final speed.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
A 0.60 kg ball falls from rest through a height of 2.0 m. Air resistance is negligible and g = 9.8 N kg⁻¹. Find its speed just before it lands.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
A van cruises at a steady 20 m s⁻¹ against a total resistive force of 600 N. (a) Find the useful power the van delivers to overcome drag. (b) The engine is supplied with 18 kW of power. Find its efficiency.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
🧠 Quick self-check
Tap each card to reveal the answer.
A force acts at 90° to the motion — how much work does it do? Zero — cos 90° = 0, so W = Fs cos θ = 0 (e.g. the normal force on a sliding block).
You double an object's speed — what happens to its kinetic energy? It becomes four times as big, because the speed is squared (2² = 4).
Does a falling object's landing speed depend on its mass? No — in mgΔh = ½mv² the mass cancels, so heavy and light objects reach the same speed (no air resistance).
How does the energy stored in a spring change if you stretch it twice as far? It becomes four times as big — EH = ½kΔx² has the stretch squared.
What is the power of a vehicle cruising at constant speed? P = Fv, where F is the resistive (drag) force the engine has to balance.
Where does the 'wasted' energy in a machine usually go? Almost always to thermal energy (heat) — it isn't destroyed, just spread out and made useless.
🎯 Exam tips
Exam Tips
- Pick the equation from what you're GIVEN: force × distance → W = Fs cos θ; mass & speed → Eₖ = ½mv²; height → ΔEₚ = mgΔh; spring stretch → EH = ½kΔx²; energy & time (or steady-speed force) → P; useful vs total → η.
- Area under a force–distance graph = the work done — a rectangle for a flat line, a triangle for a sloping spring line. The area is energy in joules, not a speed.
- Always square the variable that's squared: v in ½mv² and Δx in ½kΔx². Double it → four times the energy (the classic 'stopping distance' point).
- Conservation of energy: set PE lost = KE gained → mgΔh = ½mv²; the mass cancels, so don't carry it through.
- Slide-to-rest against friction: friction force × distance = Eₖ, so distance = Eₖ ÷ friction force.
- At constant speed use P = Fv with F as the resistive force; convert kW ↔ W and cm ↔ m before substituting.
- Efficiency is a fraction ≤ 1 — if you get more than 100%, you've divided total by useful instead of useful by total.