Power: How Fast Work Is Done – Work, Power and Energy (Senior 1 Physics) | Ganzaa

Work, Power and Energy

Lessons

Senior 1 · Physics

Work, Power and Energy

5 lessons

Lesson 2 of 5

Power: How Fast Work Is Done

22 min

Same job, different speed

Two pupils carry identical 20 kg bags of maize up the same stairs to the school store. Mugisha takes 30 seconds; Habimana takes a full minute. Both lifted the same weight through the same height, so both did the same work. Yet everyone would say Mugisha is more powerful. A motorcycle and a lorry can both climb from Nyabugogo to the city centre; the lorry, with its bigger engine, can do it with a far bigger load in the same time. Power measures how quickly work is done.

Definition

Power ( $P$ ) is the rate of doing work, or the rate at which energy is transferred:

P = \frac{W}{t}

where $W$ is the work done (or energy transferred) in joules and $t$ is the time taken in seconds.

The SI unit of power is the watt (W): 1 W is a rate of working of one joule per second, $1\,\mathrm{W} = 1\,\mathrm{J\,s^{-1}}$ . Larger units: $1\,\mathrm{kW} = 1000\,\mathrm{W}$ and $1\,\mathrm{MW} = 10^{6}\,\mathrm{W}$ . An older unit, the horsepower, is still used for engines: $1\,\mathrm{hp} \approx 746\,\mathrm{W}$ .

Warning

Two different Ws. In $P = \frac{W}{t}$ the letter $W$ stands for work, in joules. The unit symbol W stands for watt. Write units in full words when there is any risk of mixing them up.

Experiment: how powerful are you?

You need: a flight of stairs, a metre rule, a stopwatch, a bathroom scale and a partner.

Find your mass on the scale, for example 50 kg.
Measure the height of one step with the metre rule and count the steps. The vertical height is the height of one step times the number of steps.
Your partner times you as you run up the stairs from a standing start.
Work done against gravity: $W = m g h$ . Your power: $P = \frac{W}{t}$ .

Figure 2.1: The work done depends only on the vertical height climbed, 3.0 m, not on the length of the stairs.

Pupil	Mass (kg)	Height (m)	Work $m g h$ (J)	Time (s)	Power (W)
Mutesi	50	3.0	1470	4.0	368
Gatete	60	3.0	1764	6.0	294

Gatete did more work, because he is heavier, but Mutesi developed more power, because she did her work faster.

Important

Safety. Walk briskly rather than sprint if the stairs are narrow or wet, hold the rail, and never run on stairs while others are using them. Anyone with asthma or a heart condition should do the timing instead.

Worked example 1: Mutesi's power

Mutesi (50 kg) climbs a vertical height of 3.0 m in 4.0 s. Find her power.

W = mgh = 50 \times 9.8 \times 3.0 = 1470\,\mathrm{J}

P = \frac{W}{t} = \frac{1470\,\mathrm{J}}{4.0\,\mathrm{s}} = 367.5\,\mathrm{W} \approx 370\,\mathrm{W}

Worked example 2: a water pump for a school tank

A pump lifts 500 L of water (mass 500 kg) from a well to a tank 10 m higher in 5.0 minutes. Find its useful power output.

W = mgh = 500 \times 9.8 \times 10 = 49\,000\,\mathrm{J}

t = 5.0 \times 60 = 300\,\mathrm{s}, \qquad P = \frac{49\,000\,\mathrm{J}}{300\,\mathrm{s}} \approx 163\,\mathrm{W}

Power ratings at home

Every electrical appliance is labelled with its power: the energy it transfers each second. The same energy idea rates power stations.

Device	Typical power	Energy transferred in 1 minute
LED bulb	9 W	540 J
Phone charger	10 W	600 J
Television	80 W	4800 J
Refrigerator	150 W	9000 J
Electric iron	1000 W = 1 kW	60 000 J
Electric kettle	2000 W = 2 kW	120 000 J
Nyabarongo I hydropower plant	28 MW	$1.68 \times 10^{9}$ J

Definition

The kilowatt-hour (kWh) is the energy transferred by a 1 kW appliance in 1 hour. It is the “unit” on an electricity meter and on a prepaid Cash Power token from REG.

1\,\mathrm{kWh} = 1000\,\mathrm{W} \times 3600\,\mathrm{s} = 3.6 \times 10^{6}\,\mathrm{J} = 3.6\,\mathrm{MJ}

Worked example 3: energy used at home

(a) How much energy does a 2000 W kettle transfer in 3.0 minutes? (b) How many kWh does an 80 W television use in 5 hours?

\text{(a)}\quad E = Pt = 2000\,\mathrm{W} \times 180\,\mathrm{s} = 360\,000\,\mathrm{J} = 360\,\mathrm{kJ}

\text{(b)}\quad E = 0.080\,\mathrm{kW} \times 5\,\mathrm{h} = 0.40\,\mathrm{kWh}

Warning

Convert the time. For power in watts, time must be in seconds: 5.0 minutes is 300 s, not 5. For energy in kWh, use kilowatts and hours together. Never mix watts with hours unless you want watt-hours.

Tip

Rearrange the power triangle: $P = \frac{W}{t}$ , $W = P t$ , $t = \frac{W}{P}$ . A useful check: a person working hard for a long time produces roughly 100 W; a sprint up the stairs, several hundred watts.

Power (Khan Academy, work and energy series). Watch for power as work divided by time, and for why the watt is a joule per second.

✦

Practice

Define power and give its SI unit.
A machine does 1200 J of work in 4.0 s. Find its power.
Convert (a) 2.5 kW to watts, (b) 450 W to kilowatts, (c) 3 MW to watts.
A 60 W bulb is switched on for 10 minutes. How much energy does it transfer?
A 45 kg pupil climbs stairs of vertical height 4.0 m in 6.0 s. Find her power.
How long does a 1500 W motor take to do 90 000 J of work?
A crane lifts a 200 kg load through 15 m in 20 s. Find the power it develops.
A family's fridge (150 W) runs for 10 hours a day. How many kWh does it use in 30 days?

Show answersHide answers

Power is the rate of doing work (or of transferring energy): $P = \frac{W}{t}$ . SI unit: the watt, $1\,\mathrm{W} = 1\,\mathrm{J\,s^{-1}}$ .
$P = \frac{1200}{4.0} = 300\,\mathrm{W}$
(a) 2500 W; (b) 0.45 kW; (c) $3 \times 10^{6}$ W
$E = Pt = 60 \times 600 = 36\,000\,\mathrm{J}$
$W = 45 \times 9.8 \times 4.0 = 1764\,\mathrm{J}$ ; $P = \frac{1764}{6.0} = 294\,\mathrm{W}$
$t = \frac{W}{P} = \frac{90\,000}{1500} = 60\,\mathrm{s}$
$W = 200 \times 9.8 \times 15 = 29\,400\,\mathrm{J}$ ; $P = \frac{29\,400}{20} = 1470\,\mathrm{W}$
$0.150\,\mathrm{kW} \times 10\,\mathrm{h} \times 30 = 45\,\mathrm{kWh}$

Key points

Power is the rate of doing work or transferring energy: $P = \frac{W}{t}$ .
The watt: $1\,\mathrm{W} = 1\,\mathrm{J\,s^{-1}}$ ; $1\,\mathrm{kW} = 1000\,\mathrm{W}$ .
The same work done in less time means more power.
Electricity is sold in kilowatt-hours: $1\,\mathrm{kWh} = 3.6\,\mathrm{MJ}$ .

Sources and credits

7.7: Power, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Video: Power | Work and energy | Physics | Khan Academy, Khan Academy (khanacademymedicine channel), embedded from YouTube.

Written by Ganzaa: the lesson text, the experiment and its results, the diagram, the power-ratings table, worked examples and practice exercises. The Nyabarongo I capacity (28 MW) is from Rwanda Energy Group.

1Same job, different speed

2Experiment: how powerful are you?

3Power ratings at home

Same job, different speed

Experiment: how powerful are you?

Power ratings at home