Chapter 4 of 8

All 8 chapters
  1. 1Calculating energy stores, readunderstood
  2. 2Springs and elastic energy, readunderstood
  3. 3Work done, readunderstood
  4. 4Power, readunderstood
  5. 5Efficiency, readunderstood
  6. 6Specific heat capacity, readunderstood
  7. 7Energy resources, readunderstood
  8. 8Energy at home, readunderstood

S4.9.4 Chapter 4

Power

A motor transfers 200 J in 8 s. What is its power?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

  • You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Calculate power in watts from energy and time
  • Measure your own power by timing yourself climbing stairs

The same person carries a bag up the same stairs twice, once walking and once running. The energy transferred is the same, but the run is faster. Power measures how fast energy is transferred.

power = energy transferred ÷ time taken. In symbols, P = E ÷ t. Energy E is in joules (J). Time t is in seconds (s). Power P is in watts (W).

One watt is one joule each second. So a 5 W device transfers 5 J every second. A kilowatt (kW) is 1 000 W. Work done is energy transferred, so P = W ÷ t as well.

A table of power units with the columns Unit and Meaning. W means 1 J each second. kW means 1 000 W.UnitMeaningW1 J each secondkW1 000 W
The two units of power you need.

Worked example

A motor transfers 480 J in 12 s. What is its power?

  1. P = E ÷ t.
  2. P = 480 ÷ 12 = 40 W.

Answer: 40 W

Here is how power compares across things you know. These are typical sizes. Real ones differ from one make to another.

A table of about how much power some things use, with the columns Thing and Power. A resting body 80 W. An LED lamp 8 W. A television 100 W. A toaster 1 kW. An electric kettle 2 kW.ThingPowerresting body80 WLED lamp8 Wtelevision100 Wtoaster1 kWelectric kettle2 kW
About how much power each uses. Heating things takes much more power than lighting.

A graph shows power as steepness. Plot the energy transferred against time. A machine with a power of 40 W adds 40 J every second. A 10 W machine adds only 10 J every second, so its line is less steep.

A graph of energy transferred in J against time in s. The axes are labelled time (s), from 0 to 10 in steps of 2, and energy (J), from 0 to 400 in steps of 100. Two straight lines start at 0. The steeper line is labelled A 40 W and reaches 400 J at 10 s. The other is labelled B 10 W and reaches 100 J at 10 s.A 40 WB 10 W2468101002003004000time (s)energy (J)
The steeper line is the machine with more power. In 10 s, A transfers 400 J and B transfers 100 J.

Before you use P = E ÷ t, change the units. Time must be in seconds: 1 minute is 60 s. Power must be in watts: 1 kW is 1 000 W. Skip this step and the answer is wrong by 60 or by 1 000.

You can turn the sum around. energy transferred = power × time, so E = P × t. And time = energy ÷ power.

Worked example

An electric kettle has a power of 2 kW. It runs for 3 minutes. How much energy does it transfer?

  1. E = P × t.
  2. 2 kW = 2 000 W, and 3 minutes = 3 × 60 = 180 s.
  3. E = 2 000 × 180 = 360 000 J = 360 kJ.

Answer: 360 kJ

Now use your own body. When you climb stairs, you lift your own mass. The useful energy you transfer is your gain in gravitational potential energy, m × g × h. Divide it by the time and you have your useful power. Your body uses more, because much is wasted as heat. So this is the least power you made.

A person of 50 kg at the foot of a staircase. A line under the stairs is labelled 18 steps, each 18 cm high. The person is labelled 50 kg. The top is labelled climbs in 12 s.50 kgclimbs in 12 s18 steps, each 18 cm high
A person of 50 kg climbs 18 steps of 18 cm in 12 s.

Worked example

A person of 50 kg climbs 18 steps, each 18 cm high, in 12 s. What is her power? Use g = 9.8 N/kg.

  1. The useful energy she transfers is her gain in gravitational potential energy, m × g × h.
  2. The height is 18 × 0.18 = 3.24 m.
  3. E = 50 × 9.8 × 3.24 = 1 587.6 J.
  4. P = E ÷ t = 1 587.6 ÷ 12 = 132.3 W.

Answer: 132.3 W

A wrong idea many people have

A more powerful machine always does more work.

Why it feels right: In daily talk, a powerful engine means a strong one that gets more done.

The test that breaks it: A 100 W lamp on for 1 s transfers 100 J. A 10 W lamp on for 100 s transfers 10 × 100 = 1 000 J. The weaker lamp has transferred ten times as much energy.

What is true: Power is the rate: energy per second. A powerful machine does more work in the same time. A weaker one can do more work if it runs for longer. Energy = power × time.

Do not mix up the units. A joule is an amount of energy. A watt is a rate: joules each second. A lamp does not use 40 W. It uses energy at a rate of 40 W, which is 40 J every second.

Try it

Measure your power on the stairs

You need: a flight of stairs, a tape measure or ruler, a watch or phone timer, a friend to time you, your mass in kg, from a scale.

Be safe: Use stairs with a rail and clear, dry steps. Walk fast but do not run. Stop at once if you feel pain or dizzy. If you have been told not to exercise hard, do not try this.

  1. Count the steps. Measure the height of one step in cm. Work out the total height in m.
  2. Find your mass in kg. If you have no scale, use 50 kg to practise the sum.
  3. Stand at the bottom. Your friend starts the timer when you start. Walk up briskly, one step at a time, holding the rail. Walk back down slowly.
  4. Your friend stops the timer when you reach the top. Write the time in s.
  5. Work out the energy: mass × 9.8 × height. Divide by the time. That is your power in W.

What you will see: For the example below, a person of 60 kg climbs 16 steps of 17 cm in 10 s. The height is 2.72 m. The energy is 60 × 9.8 × 2.72 ≈ 1 599 J. The power is about 160 W. Try it again more slowly. The energy is the same and the time is longer, so your power is smaller.

If you cannot try it, this is what happens.
A person of 60 kg at the foot of a staircase. A line under the stairs is labelled 16 steps, each 17 cm high. The person is labelled 60 kg. The top is labelled climbs in 10 s. A line beside the stairs is labelled 2.72 m.60 kgclimbs in 10 s2.72 m16 steps, each 17 cm high

Key words

power
the energy transferred each second, P = E ÷ t.
watt
the unit of power, written W. One watt is one joule each second.
kilowatt
a unit of power, written kW. One kilowatt is 1 000 watts.

Back to your guess

The answer is D: 25 W

Power is energy ÷ time: 200 J ÷ 8 s = 25 W. Multiplying or subtracting the numbers does not give a rate.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A machine transfers 120 J of energy in 6 s. What is its power?

    Tap your answer.

    1. Yes, that is right. P = E ÷ t = 120 ÷ 6 = 20 W.

    2. Not this one. You multiplied the energy by the time. Power is energy divided by time. Try another.

    3. Not this one. You divided the time by the energy, which is upside down. Power is energy ÷ time. Try another.

    4. Not this one. You took the time from the energy. Power is a rate, so divide. Try another.

    See the answer

    20 W

    P = E ÷ t = 120 ÷ 6 = 20 W.

    • If you chose B: You multiplied the energy by the time. Power is energy divided by time.
    • If you chose C: You divided the time by the energy, which is upside down. Power is energy ÷ time.
    • If you chose D: You took the time from the energy. Power is a rate, so divide.

  2. A motor transfers 900 J of energy in 5 minutes. What is its power?

    Tap your answer.

    1. Not this one. You divided by 5, but the time must be in seconds. 5 minutes is 300 s. Go to M2.6.7. Try another.

    2. Not this one. You multiplied the energy by the time. Power is energy divided by time. Try another.

    3. Yes, that is right. 5 minutes = 5 × 60 = 300 s. P = E ÷ t = 900 ÷ 300 = 3 W.

    4. Not this one. You took the time from the energy. Power is a rate, so divide. Try another.

    See the answer

    3 W

    5 minutes = 5 × 60 = 300 s. P = E ÷ t = 900 ÷ 300 = 3 W.

    • If you chose A: You divided by 5, but the time must be in seconds. 5 minutes is 300 s. Go to M2.6.7.
    • If you chose B: You multiplied the energy by the time. Power is energy divided by time.
    • If you chose D: You took the time from the energy. Power is a rate, so divide.

  3. A lamp has a power of 0.06 kW. How many joules does it transfer each second?

    Tap your answer.

    1. Not this one. You read the kilowatts as watts. 0.06 kW is 0.06 × 1 000 = 60 W. Try another.

    2. Yes, that is right. 0.06 kW = 0.06 × 1 000 = 60 W. One watt is one joule each second, so the lamp transfers 60 J each second.

    3. Not this one. You moved the decimal point two places. A kilowatt is 1 000 W, which is three places. Try another.

    4. Not this one. You moved the decimal point by four places. A kilowatt is 1 000 W, which is three places. Try another.

    See the answer

    60 J

    0.06 kW = 0.06 × 1 000 = 60 W. One watt is one joule each second, so the lamp transfers 60 J each second.

    • If you chose A: You read the kilowatts as watts. 0.06 kW is 0.06 × 1 000 = 60 W.
    • If you chose C: You moved the decimal point two places. A kilowatt is 1 000 W, which is three places.
    • If you chose D: You moved the decimal point by four places. A kilowatt is 1 000 W, which is three places.

  4. The graph shows two machines, each doing 500 J of work. Machine A takes 10 s. Machine B takes 20 s. Which statement is true?

    A graph of energy transferred in J against time in s. The axes are labelled time (s), from 0 to 20 in steps of 5, and energy (J), from 0 to 500 in steps of 100. Two straight lines start at 0 and both reach 500 J. The steeper line is labelled machine A and the other is labelled machine B.machine Amachine B51015201002003004005000time (s)energy (J)
    Tap your answer.

    1. Not this one. B takes twice as long for the same work, so B has half the power of A. Try another.

    2. Not this one. The work is the same, but the times differ. Power is work divided by time. Try another.

    3. Not this one. Both do 500 J. B takes longer, but does not do more. Try another.

    4. Yes, that is right. A transfers 500 J in 10 s, so P = 50 W. B transfers 500 J in 20 s, so P = 25 W. A is twice as powerful.

    See the answer

    Machine A has twice the power of machine B.

    A transfers 500 J in 10 s, so P = 50 W. B transfers 500 J in 20 s, so P = 25 W. A is twice as powerful.

    • If you chose A: B takes twice as long for the same work, so B has half the power of A.
    • If you chose B: The work is the same, but the times differ. Power is work divided by time.
    • If you chose C: Both do 500 J. B takes longer, but does not do more.

  5. A lamp has a power of 5 W and is on for 2 minutes. How much energy does it transfer?

    Tap your answer.

    1. Not this one. You multiplied by 2, but the time must be in seconds. 2 minutes is 120 s. Go to M2.6.7. Try another.

    2. Not this one. You divided the power by the time. Energy = power × time, so multiply. Try another.

    3. Yes, that is right. 2 minutes = 120 s. E = P × t = 5 × 120 = 600 J.

    4. Not this one. You used 60 s, which is one minute. The lamp is on for 2 minutes, which is 120 s. Try another.

    See the answer

    600 J

    2 minutes = 120 s. E = P × t = 5 × 120 = 600 J.

    • If you chose A: You multiplied by 2, but the time must be in seconds. 2 minutes is 120 s. Go to M2.6.7.
    • If you chose B: You divided the power by the time. Energy = power × time, so multiply.
    • If you chose D: You used 60 s, which is one minute. The lamp is on for 2 minutes, which is 120 s.

  6. Ana and Tariq each have a mass of 60 kg. They climb the same stairs. Ana takes 10 s and Tariq takes 15 s. Who has the greater power, and why?

    Tap your answer.

    1. Yes, that is right. Both gain the same gravitational potential energy. Ana does it in 10 s and Tariq in 15 s, so Ana has the greater power.

    2. Not this one. Working longer does not mean more power. The same energy over more time is a smaller rate. Try another.

    3. Not this one. They lift the same mass the same height, so they transfer the same energy. Try another.

    4. Not this one. The energy is the same, but power is energy ÷ time, and their times differ. Try another.

    See the answer

    Ana, because she transfers the same energy in less time.

    Both gain the same gravitational potential energy. Ana does it in 10 s and Tariq in 15 s, so Ana has the greater power.

    • If you chose B: Working longer does not mean more power. The same energy over more time is a smaller rate.
    • If you chose C: They lift the same mass the same height, so they transfer the same energy.
    • If you chose D: The energy is the same, but power is energy ÷ time, and their times differ.

  7. From an earlier chapter

    A force of 15 N moves a box 8 m in the direction of the force. How much work is done?

    Tap your answer.

    1. Not this one. You added the force and the distance. Work done is force × distance. Go to S4.9.3. Try another.

    2. Not this one. You divided the force by the distance. Work done is force × distance, so multiply. Go to S4.9.3. Try another.

    3. Yes, that is right. W = F × s = 15 × 8 = 120 J.

    See the answer

    120 J

    W = F × s = 15 × 8 = 120 J.

    • If you chose A: You added the force and the distance. Work done is force × distance. Go to S4.9.3.
    • If you chose B: You divided the force by the distance. Work done is force × distance, so multiply. Go to S4.9.3.

  8. A machine transfers 360 J in 8 s. What is its power, in watts?

    See the answer

    45 W

    P = E ÷ t = 360 ÷ 8 = 45 W.

  9. A heater has a power of 1.5 kW and runs for 2 minutes. How much energy does it transfer, in kJ?

    See the answer

    180 kJ

    1.5 kW = 1 500 W and 2 minutes = 120 s. E = P × t = 1 500 × 120 = 180 000 J = 180 kJ.

  10. A person of 70 kg climbs 20 steps, each 15 cm high, in 10 s. Use g = 9.8 N/kg. What is her power, in watts?

    See the answer

    205.8 W

    Height = 20 × 0.15 = 3 m. E = 70 × 9.8 × 3 = 2 058 J. P = 2 058 ÷ 10 = 205.8 W.

Keep it

Copy this into your exercise book.

P = E ÷ t. Watts: 1 W is 1 J each second. Seconds, not minutes. Watts, not kW: 1 kW is 1 000 W. E = P × t. Stairs: P = m × g × h ÷ t.

Draw: A flight of stairs with its height marked, and P = E ÷ t beside it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (758 KB). It opens on any phone.