Power and Efficiency

The question

Two cranes lift the same crate to the same rooftop, doing identical work. One takes 6 seconds, the other takes 18. The first crane's motor is three times bigger and burns three times the fuel. If the work is the same, what did the bigger crane actually buy you?

Work tells you how much energy moved, but not how fast. Power is the rate of energy transfer — and it's what you pay for in horsepower, watts, and electricity bills.

The key idea

Power is the rate of energy transfer — how fast work is done — measured in watts $(1 W = 1 J/s)$. Efficiency is the fraction of input energy that comes out as useful work; the rest is lost, usually as heat. Two machines can do exactly the same work yet have very different power and very different efficiency.

Power answers 'how fast?', not 'how much?'. Because $P = W/t$, doing the same job in half the time needs twice the power. When force pushes something moving at a steady speed, $W = F\cdot d$ and $d = v\cdot t$, so $P = F\cdot d/t = F\cdot v$ — a direct link between force, speed, and power (this is why a car needs far more engine power to hold high speed than low speed). No real machine delivers all its input as useful work: efficiency = useful out ÷ total in, and the shortfall escapes as heat, sound, or friction. Back to the hook — both cranes do the same work, so the bigger crane didn't buy you more work; it bought you more POWER, finishing in 6 s instead of 18 s by transferring energy three times faster. **Limiting case:** cruising at constant speed on the flat, all engine power fights drag: $P = Fv$ with $a = 0$ — double the speed and (with $F \propto v^2$ drag) the demanded power grows eightfold. **Connect it:** $P = W/t = Fv$ is the work–energy theorem per second; a power rating is just how fast a machine can move joules.

The formula

• ·P = power in watts (W)
• ·W = work or energy transferred in joules (J)
• ·t = time in seconds (s)
• ·F = force in newtons (N)
• ·v = steady speed in metres per second (m/s). One watt is one joule delivered every second. Efficiency is a unitless ratio
• ·written as a percentage.

What to remember

• ·Power = energy transferred ÷ time = work/t, in watts.
• ·At steady speed, P = F × v.
• ·Efficiency = useful energy out ÷ total energy in; the rest is usually lost as heat.