Applied Maths: Work, Power, Energy and Momentum (Strand 5) — Higher Level Notes
Work as force along the displacement, power as tractive effort times speed, conservation of energy, drag and terminal velocity, and conservation of momentum.
1. Work: force along the displacement
W = Fs or, at an angle, W = Fs cos A
The cos A is the whole idea. Only the component of the force along the direction of motion does any work.
So a porter carrying a suitcase 30 m along a level corridor does no work at all with the supporting force: it acts vertically, the motion is horizontal, and cos 90 = 0. Multiplying 18 kg by g by 30 m is the standard wrong answer.
2. Power
P = W/t and P = Tv
Power is work per unit time, so its unit is the joule per second — the watt. One newton metre is a joule, which is work, not power. That distinction is a favourite short question.
The second form, power equals tractive effort times speed, is what makes engine problems tractable.
3. Energy, and what conservation actually says
- Kinetic energy: KE = ½mv2. It depends on the square of the speed, so doubling the speed quadruples the energy.
- Potential energy: PE = mgh.
For a freely falling ball, the kinetic energy gained is exactly the potential energy lost, so the total stays constant. Nothing is created.
Watch out — "the energy is destroyed"
When brakes stop a car, the kinetic energy is converted, mainly into heat in the brakes, tyres and road. When a pendulum swings down in air, its mechanical energy passes to the air as heat. Energy leaving the system you are studying is not the same as energy ceasing to exist, and examiners mark the difference.
4. Drag and terminal velocity
A resistive force is usually modelled as D = kvn — it grows with speed. Two standard situations follow:
- Maximum speed of a vehicle. Top speed is reached when the driving force has grown equal to the total resistance, so the acceleration falls to zero. The engine has not switched off; the forces have simply come into balance.
- Terminal velocity of a falling body. Reached when kvn = mg.
5. Conservation of momentum
m1u1 + m2u2 = m1v1 + m2v2
Momentum is conserved in every collision. Kinetic energy is not — it survives only when the collision is perfectly elastic. Assuming both are conserved is the single most costly misconception in this chapter and the next.
Momentum is a vector, so in two dimensions you conserve it separately in each direction. Signs matter: choose a positive direction and stay with it.
Pumps, and doing two jobs at once
A pump raising mass m of water per second through height h and delivering it at speed v does two things: it lifts the water (mgh) and it sets it moving (½mv2). The power is the sum, mgh + ½mv2. Answering mgh alone is right only for a pump whose water emerges with negligible speed.
Practise this chapter
QuizPerCard has Work, Power, Energy and Momentum as concept cards, a formula reference and multiple-choice practice — every question with the full derivation, and an explanation of why each wrong option is wrong. Open the practice.