PhysicsLesson 13 of 18About 30 minutes

Power

Walk up a flight of stairs or run up it: you end up at the same height, so you do the same work against gravity. Running still feels harder. The difference is power, the rate at which work is done.

By the end of this lesson you can

relate energy, time and power;
calculate power with $P = W/\Delta t$ and with $P = Fv$;
compare machines by their power ratings;
describe what $E = mc^2$ says about mass and energy.

Power is a rate

Think first

Twins of equal mass race up the same staircase. One strolls and takes 10 s; the other sprints and takes 5 s. Which twin does more work against gravity, and which twin's power is higher?

Compare your answer

They do the same work, $mgh$, because they have the same mass and rise the same height. The runner does it in half the time, so the runner's power is twice as large.

Power

$$P = \frac{W}{\Delta t}$$

Power tells you how quickly work gets done, or more generally how many joules of energy are transferred each second. The SI unit is the watt (W): $1\ \text{W} = 1\ \text{J/s}$.

A light bulb rated at 10 W converts 10 joules of electrical energy every second. Engines are often rated in horsepower: $1\ \text{hp} \approx 746\ \text{W}$. Be careful not to confuse W (watt, the unit) with $W$ (work, the quantity).

Electricity bills use the kilowatt-hour, which is a unit of energy, not power: one kilowatt delivered for one hour, $1\ \text{kWh} = (1000\ \text{W})(3600\ \text{s}) = 3.6 \times 10^6\ \text{J}$.

Choosing a motor

For a given job, the work is fixed. A more powerful machine does the same work in less time: $\Delta t = W/P$. Engineers choose a motor that is powerful enough for the time they need, without wasting money or energy on one that is far too big.

Three motors, one job

Interactive

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

Each motor lifts the same 150 kg stage set by 6.0 m. Is the work the same for all three?
Predict which motor takes closest to 4.0 s, then press Lift the curtains to check.

Worked example 1

A theater crew wants to raise a 150 kg piece of scenery 6.0 m at a steady speed, and the lift should take about 4.0 s. Motors rated at 1.5 kW, 2.5 kW and 4.0 kW are available. Which is the best choice?

Work needed. At constant speed the lifting force equals the weight: $W = mgh = (150)(9.81)(6.0) = 8.8 \times 10^3\ \text{J}$.
Time for each motor. $\Delta t = W/P$: $$\frac{8830}{1500} = 5.9\ \text{s},\qquad \frac{8830}{2500} = 3.5\ \text{s},\qquad \frac{8830}{4000} = 2.2\ \text{s}$$
Compare with 4.0 s.
The 2.5 kW motor (3.5 s) comes closest.

Power, force and speed

When a constant force pushes an object along at speed $v$, the work done in time $\Delta t$ is $W = Fd = F(v\Delta t)$. Divide by $\Delta t$:

Power from force and speed

$$P = Fv$$

Use it when the force is along the motion. If the speed changes, this gives the power at each moment.

This explains why a car engine works harder at high speed. To cruise at constant velocity, the engine must supply a forward force equal to the air resistance and rolling friction. Air resistance grows quickly with speed, and $P = Fv$ multiplies that larger force by the larger speed.

Worked example 2

A car cruises at a constant 30.0 m/s on a level highway. Air resistance and rolling friction together push back with 600 N. What power must the engine deliver to the wheels?

Constant velocity, so the forward force equals the resistance: $F = 600\ \text{N}$.
Use $P = Fv$. $P = (600\ \text{N})(30.0\ \text{m/s}) = 1.80 \times 10^4\ \text{W}$
18.0 kW, about 24 hp

Worked example 3

An 800 kg elevator carries 400 kg of passengers upward at a constant 3.00 m/s. A friction force of 2000 N opposes its motion. What is the minimum power of the motor?

Total mass. $800 + 400 = 1200\ \text{kg}$, so the weight is $(1200)(9.81) = 1.18 \times 10^4\ \text{N}$.
At steady speed the cable must hold up the weight and overcome friction. $F = 11\,770 + 2000 = 1.38 \times 10^4\ \text{N}$.
Power. $P = Fv = (1.38 \times 10^4)(3.00) = 4.13 \times 10^4\ \text{W}$
About 41 kW

Puzzle

Mountain roads usually zigzag up the slope instead of going straight to the top. Why does that help a car with a small engine?

Show explanation

The car gains the same height either way, so the work against gravity is the same. The zigzag road is longer and less steep, so the car can climb with a smaller force spread over a longer distance and a longer time. That needs less power at any one moment, which a small engine can supply.

Mass is a form of energy

Can you keep pushing an object faster and faster forever? In 1905 Albert Einstein showed that the answer is no. As an object approaches the speed of light, $c = 3.00 \times 10^8\ \text{m/s}$, its kinetic energy grows much faster than $\tfrac12 mv^2$ predicts, and reaching $c$ would take infinite energy. At everyday speeds the difference is far too small to notice, which is why $\tfrac12 mv^2$ works so well.

Einstein also found that mass itself is a form of energy. Even an object at rest has a rest energy:

Rest energy

$$E_R = mc^2$$

Because $c^2$ is enormous, a tiny mass holds a huge amount of energy. Nuclear power stations and the Sun release energy by converting a small fraction of mass into other forms.

Quick check

How much rest energy is locked up in a paper clip with a mass of 1.0 g?

Show solution

$E_R = (1.0 \times 10^{-3}\ \text{kg})(3.00 \times 10^8\ \text{m/s})^2 = 9.0 \times 10^{13}\ \text{J}$, roughly the energy released by 20 000 tonnes of TNT.

Check yourself

Question 1

A 60.0 kg student walks up stairs that rise 4.00 m in 5.00 s. What is her average power output against gravity?

Show solution

$P = \dfrac{mgh}{\Delta t} = \dfrac{(60.0)(9.81)(4.00)}{5.00} = 471\ \text{W}$

Question 2

How long does a 1.50 kW pump take to raise 3000 kg of water by 12.0 m?

Show solution

$W = (3000)(9.81)(12.0) = 3.53 \times 10^5\ \text{J}$. $\Delta t = \dfrac{3.53 \times 10^5}{1500} = 235\ \text{s}$, about 4 minutes.

Question 3

A 1200 kg car accelerates from rest to 15.0 m/s in 6.00 s on a level road. Ignoring friction, what average power does the engine deliver?

Show solution

The work equals the gain in kinetic energy: $W = \tfrac12 (1200)(15.0)^2 = 1.35 \times 10^5\ \text{J}$. $P = \dfrac{1.35 \times 10^5}{6.00} = 2.25 \times 10^4\ \text{W}$ (22.5 kW).

Question 4

Two horses pull a wagon at a steady 1.50 m/s, each pulling forward with 300 N. What total power do they deliver?

Show solution

Total force 600 N: $P = Fv = (600)(1.50) = 900\ \text{W}$, a little more than one horsepower in total.

Question 5

An electricity bill charges for "kilowatt-hours". What kind of quantity is a kilowatt-hour?

Energy
Power
Force
Time

Show solution

Power multiplied by time is energy: $1\ \text{kWh} = 3.6 \times 10^6\ \text{J}$.

Summary

Power measures how fast energy is transferred: $P = W/\Delta t$, in watts ($1\ \text{W} = 1\ \text{J/s}$).
For a force along the motion, $P = Fv$.
A more powerful machine does the same work in less time.
Mass is a form of energy: $E_R = mc^2$.

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