Power
1. Power is a rate
Power is how fast energy changes. That covers energy transferred into or out of a system and energy converted from one kind to another inside it. Two processes that move the same energy have the same work and very different powers if one of them takes ten times as long.
P_avg = ΔE / Δt = W / Δtin watts, which are joules per second
One watt is one joule per second. A watt is therefore a rate and not an amount, which is why a kilowatt hour, a rate multiplied by a time, comes out as an energy and not a power. That is the unit your electricity bill is written in.
2. Two students, one staircase
The staircase is the standard laboratory for power, because everything in it can be measured with a meterstick, a scale and a stopwatch. The work done against gravity is the weight times the vertical height only, so how long the staircase is along its slope never enters the answer. Run it instead of walking and the work is unchanged while the power goes up.
Two climbers, the same stairs, different clocks. Set a mass and a height, then give each one a time. Watch which number changes and which one refuses to.
3. Force times speed
For a constant force, the instantaneous power delivered is the component of the force along the velocity, multiplied by the speed.
P = F∥ v = F v cos θθ lies between the force and the velocity
Note where that angle is measured. It is between the force and the velocity, not between the force and the ground, and a figure that labels an angle to the horizontal and then drops it into this expression has changed the meaning of the symbol.
A car at steady speed is the clean case. It is not accelerating, so the driving force exactly equals the total resistance, and the power the engine delivers is that force times the speed.
A car holding a steady speed. The resistance grows with speed, which is why the power needed grows faster than the speed does. Push the speed slider and watch the last number climb out of proportion.
4. Conversion inside a system
Power is the rate of any energy change, including a conversion that happens entirely inside a system with no work done on it at all. A battery turning chemical energy into electrical energy is the case. Nothing is pushed through a distance, nothing crosses the boundary, and power is still exactly the right word.
A course that teaches power only as work over time has taught one of the two definitions and quietly skipped the other, and the skipped one is the one your phone charger runs on.
Check yourself
1. A motor does 4500 J of work in 30 s. Find its average power.
P = 4500 / 30 = 150 W. A hundred and fifty joules every second, for thirty seconds.
2. A 55 kg student climbs a 4.2 m staircase in 6.0 s, then walks the same stairs in 18 s. Find both powers and say what did not change.
W = 55 × 10 × 4.2 = 2310 J both times. Fast: 2310 / 6.0 = 385 W. Slow: 2310 / 18 ≈ 128 W. The work did not change, because neither the weight nor the height did.
3. A car holds a steady 25 m/s against a total resistive force of 600 N. Find the power the engine delivers, and say why the car is not accelerating.
P = 600 × 25 = 15 000 W, or 15 kW. It is not accelerating because the driving force equals the resistance, so the net force is zero. All the engine power is going into the air and the road.
4. A 60 W bulb runs for 5.0 hours. Find the energy used in kilowatt hours and in joules, and say whether a kilowatt hour is a power or an energy.
0.060 kW × 5.0 h = 0.30 kWh, which is 60 × 5 × 3600 = 1.08 × 10⁶ J. A kilowatt hour is an energy: a kilowatt is a rate, an hour is a time, and a rate times a time is an amount.
That is Unit 3. Work is the transfer, kinetic and potential are the accounts, conservation is the rule the accounts obey, and power is the clock. Unit 4 puts a different conserved quantity, momentum, next to this one.