Power

What this lesson covers in the specification

Incandio is aligned to this specification. It is not published by, endorsed by or affiliated with Pearson, and it reproduces none of Pearson's wording — the statement numbers are given so you can check every lesson against your own copy.

• 4.16 — Power as the rate of transfer of energy or the rate of doing work
• 4.17 — Use P = W ÷ t

1 · Understand it

No exam language yet. The only question this section answers is: do I actually understand what is happening?

Two students carry identical boxes up the same staircase. One takes ten seconds and the other takes forty. They have done exactly the same amount of work — same force, same height — and transferred exactly the same energy. What differs is the RATE, and that is POWER: the energy transferred per second, or equivalently the work done per second.

The equation is P = W ÷ t, where W is the work done or energy transferred in joules and t is the time in seconds. The unit is the WATT, and a watt is simply a joule per second. A 60 W lamp transfers 60 J every second it is switched on. The first student above was four times as powerful as the second, and neither did any more work than the other.

Think of it like the difference between a wage and a salary

Two people are each paid £30,000 for a piece of work. One takes a month over it and one takes a year. The total is identical — that is the work done — but their rate of earning differs by a factor of twelve, and the rate is what anyone would actually quote. Power is the rate. It is why a kettle is sold as 3 kW rather than by the total energy it will ever transfer, and why an engine is rated in horsepower rather than in joules. Nobody buys a machine for the total energy it can eventually deliver; they buy it for how quickly it can deliver it.

Power used three ways

(a) A crane does 90 000 J of work in 30 s. Find its power. (b) A 2.0 kW heater runs for 5.0 minutes. Find the energy transferred. (c) A 500 W motor transfers 40 000 J. Find the time taken.

1. (a) P = W ÷ t = 90 000 ÷ 30 = 3000 W, or 3.0 kW.
2. (b) Convert: 2.0 kW = 2000 W, and 5.0 minutes = 300 s. W = P × t = 2000 × 300 = 600 000 J.
3. (c) t = W ÷ P = 40 000 ÷ 500 = 80 s.
4. In every case, convert kilowatts to watts and minutes to seconds before substituting.

Answer: 3.0 kW, 600 000 J, and 80 s.

Finding the power of a person climbing stairs — a standard question

1. Find the WEIGHT of the person: W = m × g.
2. Find the WORK DONE, which is the weight multiplied by the VERTICAL height climbed: W = F × d. The distance along the staircase is not what matters.
3. That work equals the gravitational potential energy gained.
4. Divide by the TIME taken: P = W ÷ t.
5. Check the answer is plausible: a person running upstairs manages a few hundred watts, and no human sustains more than about a kilowatt for long.

That last check is worth internalising, because it makes an implausible answer visible. A person working steadily produces something like 75 W — which is why the old measure of a horse's output, defined for exactly this purpose of comparing machines with the animals they replaced, came out at about 750 W. A kettle is 3 kW, a car engine perhaps 60 kW, and a large power station a few hundred megawatts. If your answer for a person comes out at 50 kW, the arithmetic has gone wrong somewhere.

It is worth being clear how this equation relates to the electrical one from Topic 2. There, power was P = I × V; here it is P = W ÷ t. They are not rival definitions. Power is always energy per second, and the two equations are simply different ways of getting at it depending on what you have been told — a current and a voltage, or an amount of energy and a time. A 2 kW kettle is 2 kW by either route.

2 · Grade 9 Notes

A different job from the section above. You have already understood it; this is the precise set of things to LEARN — definitions to reproduce word for word, processes in order, equations with units, and the answers that score full marks.

P = W ÷ t

P is power in watts (W), W is work done or energy transferred in joules (J), t is time in SECONDS (s)

Units: W, J, s. Rearranged: W = P × t and t = W ÷ P. One watt is one joule per second.

Learn this definition · Power

The rate at which energy is transferred, or equivalently the rate of doing work. One watt is one joule of energy transferred per second.

The two power equations — one quantity, two routes to it

A. P = W ÷ t is used when you know an amount of energy transferred and the time it took.

B. P = I × V is used when you know a current and a voltage. Both give the same quantity in watts, because power is always energy per second.

Typical powers, for checking whether an answer is sensible

• A person working steadily: about 75 W — and a horse about ten times that, which is where horsepower came from
• A filament lamp: 60 W. A modern LED lamp doing the same job: under 10 W
• An electric kettle: about 3 kW
• A family car engine: around 60 kW
• A large power station: several hundred megawatts

Finding the power developed by someone climbing stairs

1. Calculate the person's weight: force = m × g.
2. Multiply by the VERTICAL height climbed, not the distance along the stairs, to get the work done.
3. Divide the work done by the time taken in seconds.
4. Check the answer against a plausible human output of a few hundred watts.

Work done against power — a total and a rate

A. WORK DONE is a TOTAL, in joules. Two people carrying identical loads up the same stairs do identical work, however long they take.

B. POWER is a RATE, in watts. The one who takes a quarter of the time is four times as powerful, having done exactly the same work.

Model answer [3 marks]

A motor does 24 000 J of work in 40 s. Calculate its power. [3]

Using P = W ÷ t, the power is 24 000 ÷ 40 = 600 W.

Model answer [5 marks]

A 55 kg student runs up a staircase of vertical height 3.2 m in 4.0 s. Take g as 10 N/kg. Calculate the power they develop. [5]

The student's weight is m × g = 55 × 10 = 550 N. The work done is the weight multiplied by the vertical height: 550 × 3.2 = 1760 J. The power is the work done divided by the time: P = 1760 ÷ 4.0 = 440 W. This is a plausible figure for a person running hard for a few seconds.

Not this: A more powerful machine does more work than a less powerful one.

This: It does the same work FASTER. Power is a rate, not a total — two machines can do identical work while differing enormously in power, and the difference is entirely in the time taken.

Mark-losing trap. Convert kW to W and minutes to SECONDS before substituting. A watt is a joule per second.

Mark-losing trap. For stair questions use the VERTICAL height, never the distance along the staircase.

Mark-losing trap. Power is a RATE. Doing the same work in less time means more power, not more work.

Mark-losing trap. Check the answer: a person is a few hundred watts, not tens of kilowatts.

3 · Prove it — the five questions

The five questions climb Grade 6 → Grade 7 → Grade 8 → Grade 9 → Grade 9 challenge, and are marked inside Incandio on your own device, by rule, with an authored diagnosis of the mistake you actually made. The mark schemes stay in the app so that the practice is worth doing; the questions themselves are here.

1. Grade 6 · Calculate [2 marks] — A machine transfers 1200 J of energy in 4.0 s. Calculate its power.
2. Grade 7 · Explain [2 marks] — Two students carry identical boxes up the same staircase. One takes 10 s and the other takes 40 s. Compare the work done and the power developed.
3. Grade 8 · Calculate [4 marks] — A 2.5 kW heater is switched on for 3.0 minutes. Calculate the energy it transfers.
4. Grade 9 · Calculate [5 marks] — A 70 kg student runs up a staircase of vertical height 4.0 m in 5.0 s. Take g as 10 N/kg. Calculate the power they develop.
5. 9+ · Analyse [6 marks] — A shop sells two kettles. Kettle A is rated at 3.0 kW and kettle B at 1.5 kW. The salesperson says: 'Kettle A is twice as powerful, so it will use twice as much electricity to boil the same amount of water.' Select every statement that belongs in a full-mark analysis.

The people behind this science

Two ways into the same idea — the one who invented a unit of power in order to sell engines, and the one who supplied the joule that the watt is per second of. Inside Incandio each of them answers knowing exactly which lesson you have just finished.

James Watt — the one who invented a unit of power in order to sell engines

The unit on this page is his name, and he arrived at the idea of power for an entirely commercial reason. Having built an engine far better than anything available, he had to tell a mill owner what it was worth — and the thing it was replacing was horses. So he measured what a horse could actually do, in weight raised through a height in a given time, and defined HORSEPOWER as the unit of comparison. It is a rare case of a physical quantity being invented because somebody needed to price something.

• “How did you actually measure what one horse could do?”
• “Why did you need a unit of power to sell an engine?”
• “What is the difference between a powerful engine and an efficient one?”
• “How did mill owners react to being sold a machine in horses?”
• “What would you make of the unit of power carrying your name?”

James Prescott Joule — the one who supplied the joule that the watt is per second of

A watt is a joule per second, so this page depends entirely on the joule being a real, measurable quantity — and that is Joule's contribution. Until he established the exchange rate between mechanical work and heat, there was no single unit in which the output of an engine, the warming of water and the running of an electric current could all be expressed. Once there was, the rate of transfer became a meaningful number rather than a comparison between incommensurable things.

• “Why does a rate need a well-defined total before it means anything?”
• “How did you establish that work and heat could share one unit?”
• “What did engineers use to compare machines before that?”
• “Is power a more useful number than total energy?”
• “How precisely could you measure energy in your time?”

Carry on through the course

• Previous lesson: Gravitational Potential and Kinetic Energy
• Next lesson: Generating Electricity
• Energy Stores and Transfers — Where energy sits, how it moves, and why it is never used up however much it feels like it.
• Efficiency, Sankey Diagrams and Insulation — Working out what share of the energy did the job you wanted — and why the answer is never all of it.
• Conduction, Convection and Radiation — Three ways energy moves from hot to cold — and the one question that tells them apart.
• Work Done — The precise meaning of a word everybody thinks they already know — and why holding something still is not work at all.
• Gravitational Potential and Kinetic Energy — Two equations, one conversion — and why the square in the second one changes everything.
• All of Physics · Physics key terms · required practicals · equations · Incandio Science · The 4PH1 specification