Power

Two people each carry a 20 kg box up the same flight of stairs. One sprints up in 5 seconds; the other plods up in 30 seconds. They both did the same work — same mass, same height — but most people would agree the sprinter “worked harder” in some sense.

That intuition is what physics calls power: the rate at which work is done. Power isn’t about how much total energy you transfer — it’s about how quickly. The sprinter delivered the same energy in $\frac{1}{6}$ the time, so their power output was 6× higher.

Once you separate “energy” from “energy per second,” a lot of everyday things click. A 100 W bulb and a 60 W bulb both convert electrical energy into light, but the 100 W bulb does it faster — that’s why it’s brighter. A small car engine and a sports car engine can both move the cars forward, but the sports car engine can deliver that energy faster — so it accelerates harder.

The Power Equations

The second formula isn’t a new concept — it’s just an algebraic shortcut. Combine $P = W/t$ with $W = Fd$: $P = Fd/t = F(d/t) = Fv$. Same physics, different inputs. Use whichever the problem makes easy.

Units of Power

For quick estimation: 1 hp ≈ 750 W, so 60 hp ≈ 45 kW. A typical microwave is ~1000 W = 1 kW. A typical car engine peaks at ~150 hp ≈ 110 kW. A fit human can sustain ~200–400 W indefinitely (and briefly hit much higher peaks).

Power and Energy Over Time

Since $P = W/t$, you can rearrange to $W = Pt$ — the energy transferred over a time interval equals power times time. This is exactly how electricity bills are calculated: you pay for energy in kilowatt-hours (kWh), not in watts.

$1 \text{ kWh} = (1000 \text{ W})(3600 \text{ s}) = 3{,}600{,}000 \text{ J} = 3.6 \text{ MJ}$

Run a 100 W bulb for 10 hours = 1 kWh of energy. At ~$0.15 per kWh (US average), that’s about 15 cents.

Applying $P = Fv$

A car engine has to push hard enough to overcome friction and air resistance. At a constant highway speed (no acceleration), the engine’s forward push must exactly cancel the backward resistive forces. The power the engine has to deliver is $P = Fv$, where $F$ is the resistive force and $v$ is the cruising speed.

This means a higher cruising speed needs more power, even if you assume the resistive forces stay the same. And since air resistance actually grows quickly with speed (roughly with $v^2$), the power needed to cruise at 80 mph is several times the power needed at 40 mph. That’s the main reason highway driving uses way more gas than city driving (per mile of distance).

Average vs. Instantaneous Power

$P = W/t$ gives the average power over a time interval. $P = Fv$ (with instantaneous $v$) gives the instantaneous power at one specific moment.

On the MCAT, most problems are about average power, but read the question carefully — “the engine’s peak power” or “the power at the moment v = 30 m/s” both ask for instantaneous values.

Worked Example

A 70 kg person runs up a 5 m staircase in 4 seconds. What’s their average power output?

• Work against gravity: $W = mgh = (70)(10)(5) = 3500$ J.
• Power: $P = W/t = 3500/4 = 875$ W.

That’s about 1.2 hp — a serious sustained burst, but typical for a fit person sprinting upstairs. The body can briefly exceed this, but most adults can only sustain a few hundred watts for any extended period.