Two students climb the same staircase to the same landing. One sprints up it in four seconds, red-faced and panting; the other strolls up in half a minute, barely breaking a sweat. They both end up in exactly the same place, so they have both done exactly the same work against gravity — the same energy has been shifted from their muscles into lifting their body up the stairs.

So what is different? The sprinter is more powerful. They moved that energy faster — the same job packed into a much shorter time. That "how fast is energy being shifted?" is the idea of power, and it is one of the most useful numbers in all of physics. It is stamped on the side of every kettle, every light bulb, every engine you will ever meet.

Power is the rate of transferring energy

Power is the amount of energy transferred (or the amount of work done) each second. It is a rate — energy divided by the time it took.

Here E is the energy transferred (in joules, \text{J}), W is the work done (also in joules — work is energy transferred), and t is the time taken (in seconds, \text{s}). Power comes out in watts, symbol \text{W}.

A watt has a beautifully simple meaning: one watt is one joule every second.

A device rated at 100\ \text{W} is shifting 100 joules of energy every single second it is switched on. Double the power and you shift energy twice as fast; a more powerful machine does the same job in less time, or does more work in the same time.

Worked examples

The single formula P = E/t answers three kinds of question, depending on which quantity you are missing. Cover up the one you want and read it straight off the triangle E = P\,t.

Example 1 — find the power. A crane does 12\,000\ \text{J} of work lifting a steel beam, and it takes 8\ \text{s}. What is its power?

The crane transfers 1500 joules every second — that is 1.5\ \text{kW}.

Example 2 — find the energy. A kettle is rated 2\ \text{kW} = 2000\ \text{W} and boils for 90\ \text{s}. How much energy does it transfer? Rearrange to E = P\,t:

Example 3 — find the time. A 60\ \text{W} bulb has transferred 1800\ \text{J} of energy. For how long was it on? Rearrange to t = E/P:

Notice how the units keep you honest: joules divided by seconds give watts; watts times seconds give joules back. If your units don't work out, your rearrangement has gone wrong.

Everyday power: bulbs, kettles and kilowatts

Power ratings sort the world into gentle trickles and roaring torrents of energy. Look at the label on things around you and you will start to recognise the scale:

• a modern LED bulb: about 10\ \text{W};
• an old-fashioned filament bulb: about 60\ \text{W};
• a person jogging up a flight of stairs: a few hundred watts, roughly 300\ \text{W};
• a kitchen kettle: about 2000\ \text{W} = 2\ \text{kW};
• a small car engine at full tilt: tens of thousands of watts, \sim 50\ \text{kW}.

Because everyday appliances gulp thousands of watts, we bundle them into kilowatts: 1\ \text{kW} = 1000\ \text{W}. A 2\ \text{kW} kettle is more powerful than thirty old 60\ \text{W} bulbs put together — which is exactly why it can boil water in seconds while a bulb would take all day to warm the same cup.

Lifting things: P = \dfrac{mgh}{t}

A very common power question is about lifting. When you raise an object of mass m straight up by a height h, the work done against gravity is W = mgh (its gravitational potential energy goes up by that much). Do it in a time t and the power is simply that work divided by the time:

Worked example. A student of mass 50\ \text{kg} runs up a staircase that rises 4\ \text{m}, and does it in 8\ \text{s}. Taking g \approx 10\ \text{N/kg}, the work done against gravity is

so the student's useful power output climbing the stairs is

This is exactly the kind of "run up the stairs and time yourself" experiment schools use to measure a person's power — and it is why the sprinter from the top of the page, doing the very same 2000\ \text{J} job in 4\ \text{s} instead of 8\ \text{s}, comes out at 500\ \text{W}: twice as powerful.

See it: the power meter

Below, a device transfers a fixed amount of energy E in a time t, and the bar shows the resulting power P = E/t in watts. Slide the two controls and watch two things:

• turn the energy up (more joules shifted) and the power grows;
• turn the time down (same job, done quicker) and the power grows too.

Line the bar up with the 60 W bulb and 2 kW kettle markers to feel how ordinary appliances compare. This is the heart of the idea: power is not how much energy, it is how fast you deliver it.