Work, energy and power

What you’ll learn

• Use work–energy equations including resistance.
• Relate power to force and speed.

Understand the idea

Work is the component of force along displacement multiplied by the distance moved. The work–energy principle says that total work done on a particle equals its change in kinetic energy. Gravity can instead be represented by a change in gravitational potential energy, provided it is not counted twice.

An energy equation often avoids finding acceleration or time. Include work lost against friction and resistance with a consistent sign. Power is the rate of doing work; for a driving force parallel to velocity, $P=Fv$, so a constant-power engine provides less driving force at greater speed.

Key formulae

$$W=Fs\cos\theta$$

$$E_k=\frac12mv^2,\qquad E_p=mgh$$

$$W_{\mathrm{net}}=\frac12mv^2-\frac12mu^2$$

$$P=\frac{dW}{dt}=\mathbf F\cdot\mathbf v$$

Worked example

A $2\,\mathrm{kg}$ particle starts from rest and drops through a vertical height of $5\,\mathrm m$ on a smooth track. Find its speed. Take $g=9.8\,\mathrm{m\,s^{-2}}$.

1. The lost gravitational energy is $mgh=2(9.8)(5)=98\,\mathrm J$.
2. With no resistance this becomes kinetic energy: $\frac12(2)v^2=98$.
3. Take the positive square root for speed.

Answer: $v=\sqrt{98}=9.90\,\mathrm{m\,s^{-1}}$ (3 s.f.).