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AS 5

Work, energy and power

AS Level · syllabus topic 5 · theory on Papers 1 and 2; the practical skills behind it on Paper 3

Energy methods let you skip the details of a motion and jump straight from “before” to “after”. That is their whole point, and it is why an energy solution is often three lines where a force solution is fifteen.

🎯What you need to be able to do

  • Understand work done as the product of force and displacement in the direction of the force, and use \( W = Fs\cos\theta \).
  • Recall and apply the principle of conservation of energy.
  • Derive and use \( E_k = \tfrac{1}{2}mv^{2} \), and use \( \Delta E_p = mg\Delta h \) for changes in gravitational potential energy near the Earth’s surface.
  • Understand and use the concept of efficiency, and calculate it as useful output divided by total input.
  • Define power as work done per unit time, and derive and use \( P = Fv \).

📚The physics

Work transfers energy. If the force and the displacement are not parallel, only the component of the force along the displacement does work: \( W = Fs\cos\theta \). Three consequences follow immediately. A force perpendicular to the motion does no work, which is why the tension in a string does no work on a mass whirling in a circle, and why the normal contact force on a body sliding along a floor does none. A force opposing the motion does negative work. And a force acting on a stationary body does no work no matter how large it is.

A note for later. When a gas at constant pressure \(p\) expands by \(\Delta V\), it pushes the surroundings back and does work \( p\Delta V \). That is a topic 16 objective rather than a topic 5 one, but it is the same idea of work done by a force through a distance, and it is worth meeting here.

Kinetic energy is derived, not assumed. Apply \( W = Fs \) with \( F = ma \) and \( v^{2} = u^{2} + 2as \) to a body starting from rest: \( W = mas = m(v^{2}/2s)s = \tfrac{1}{2}mv^{2} \). Being able to reproduce that derivation is itself examinable.

Potential energy. \( \Delta E_p = mg\Delta h \) is valid only where the field is effectively uniform, i.e. over heights small compared with the radius of the Earth. Topic 13 replaces it with the general expression.

Efficiency = useful output energy / total input energy, and it can never exceed 1. If a calculation gives more than 100% you have either counted an energy twice or mislabelled the input.

Power. \( P = W/t \). For a body moving at speed \(v\) against a resistive force, the driving force does work \( Fv \) per second, so \( P = Fv \). At the maximum speed of a vehicle the driving force equals the total resistive force, and the output power is a maximum — this is the standard exam scenario.

✏️Worked example

A cyclist and bicycle of total mass 85 kg free-wheel from rest down a slope, descending a vertical height of 32 m, and reach the bottom at 14 m s\(^{-1}\). Find the energy dissipated against friction and air resistance. Take \( g = 9.81 \) m s\(^{-2}\).

Loss of gravitational potential energy \( = mg\Delta h = 85 \times 9.81 \times 32 = 2.67 \times 10^{4} \) J.

Gain in kinetic energy \( = \tfrac{1}{2}mv^{2} = 0.5 \times 85 \times 14^{2} = 8.33 \times 10^{3} \) J.

\[ E_{\text{dissipated}} = 2.67 \times 10^{4} - 8.33 \times 10^{3} = 1.84 \times 10^{4}\ \text{J} \]
The mark people actually lose here is using the length of the slope instead of the vertical height. \( mg\Delta h \) needs the vertical drop; if the question gives you a slope length and an angle, you must multiply by \(\sin\theta\) first. The second most common loss is answering “the energy is lost”. It is not lost. It is transferred to the internal energy of the bearings, tyres and surrounding air, and examiners want that sentence.

🔭See it happen

Drop a bouncing ball beside a metre rule and film it. The ratio of rebound height to drop height gives you the fraction of energy returned in a single bounce, and repeating with a squash ball warmed in your hand versus one straight from a cold room makes the energy accounting visible rather than theoretical.

📝Practise

Worksheet AS 5 — to be linked.

🔗Go deeper — other people’s work

The links below are not mine. They are here because they are good, and they may move or disappear without warning.

  • PhET, Energy Skate Park — the bar chart makes the transfers explicit, including the thermal share when friction is on.
  • The Physics Classroom, “Work, Energy and Power”.
  • Isaac Physics, “Energy” problem sets.