Power

How fast is the work done?

A lift and a staircase both raise you to the third floor, doing the same work against gravity. The lift is quicker. Power measures how fast work is done:

The unit is the watt: $1\ \text{W} = 1\ \text{J/s}$. An old unit, the horsepower, is about $746\ \text{W}$.

Power and speed

A force $F$ moving its point of action at speed $v$ along itself does work $F\,\Delta x$ in time $\Delta t$, so

At a steady speed, a car's engine force just balances the rolling resistance and the air drag. The drag grows as $v^2$, so the power grows faster still. See how.

You should find that at high speed almost all the power goes into pushing air aside, and that doubling the speed needs far more than double the power.

Efficiency and the kilowatt-hour

No machine turns all its input into useful work:

Electricity bills count kilowatt-hours: $1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^{6}\ \text{J}$. It is a unit of energy, not power.

Watch out. $P = Fv$ gives the power at one instant. Over a time when the speed changes, the average power is the total work over the total time.

A worked example

A lift of mass $800\ \text{kg}$ carries four people of $65\ \text{kg}$ each up $20\ \text{m}$ at a steady speed in $25\ \text{s}$.

1. Mass lifted. $800\ \text{kg} + 4 \times 65\ \text{kg} = 1060\ \text{kg}$.
2. Work. $mgh = 1060\ \text{kg} \times 9.8\ \text{m/s}^2 \times 20\ \text{m} = 207\,760\ \text{J}$.
3. Power. $P = \dfrac{207\,760\ \text{J}}{25\ \text{s}} = 8310\ \text{W}$, about $8.3\ \text{kW}$, before any losses in the motor.

Think first. Ask what the work is for (lifting, speeding up, beating friction), then divide by the time.

Practise with new numbers

Try it in Python

The program finds the power a car needs at several steady speeds.

Remember these

• $P = \dfrac{W}{t}$, in watts; $P = Fv$ for a force at speed $v$.
• Efficiency $=$ useful out $\div$ in.
• $1\ \text{kWh} = 3.6 \times 10^{6}\ \text{J}$, an energy.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A $1.2\ \text{kW}$ kettle runs for $3.0$ minutes. How much energy does it use?

2. A cyclist pushes with $40\ \text{N}$ against drag at a steady $9.0\ \text{m/s}$. What power does she deliver?

3. A motor takes in $500\ \text{W}$ and gives out $380\ \text{W}$ of useful power. What is its efficiency, and where does the rest go?

4. Why does a car use much more fuel per kilometre at $120\ \text{km/h}$ than at $80\ \text{km/h}$?