Physics I: Mechanics › Work & energy › full formula sheet

P = W / Δt

Say it: “power equals work divided by time — how fast the energy flows”

Power

Work tells you how much energy moved; power tells you how fast. Same work, half the time, double the power.

P is power in watts, W work in joules, Δt time in seconds. 1 W = 1 J/s. (Also: P = Fv for a constant force at speed v.)

Before this lesson: Work (constant force)

Where it comes from

Two cranes lift identical 1000 kg beams to the same floor. One takes 10 s, the other takes 60 s. Same work — but nobody would call them equally powerful. Power is the rate: work divided by the time it took.

Before reading on: you climb a flight of stairs (doing ~3000 J of work against gravity) in 10 s, then again in 30 s. Which climb had more power, and by what factor?

The 10 s climb: 3× the power. Same 3000 J, one-third the time: P = 300 J/s vs 100 J/s. Your legs' effort rate tripled even though the total work matched.

P = W / Δtenergy per second — the speed of the energy flowSay it: “power equals work over time”

There's a second form you'll use constantly. If a constant force F pushes something already moving at speed v, then in time Δt the work is F·(vΔt), so:

P = Fvforce times velocity — power without the clockSay it: “power equals force times velocity”

Derivation

Both forms come from the definition of work:

Pavg
=
W / Δt
Step 1 — definition. Average power is work per unit time. This is the primary meaning of “power”.
P
=
F·Δx / Δt = F·v
Step 2 — constant force at speed v. W = FΔx (parallel), and Δx/Δt = v. The time cancels: power from force and speed directly.
P
=
dW/dt = F · v
Step 3 — instantaneous. In calculus form, power is the time derivative of work — the dot product of force and velocity vectors. ∎

How to use it

The procedure, every time:

  1. Decide: total work known, or force + speed? Work and time → P = W/Δt. Constant force at known speed → P = Fv (faster).
  2. Compute W first if needed. Lifting: W = mgh. Pushing: W = Fd. Then divide by Δt.
  3. Watch the time units. Seconds for watts. Minutes/hours must convert (or you'll be off by 60×/3600×).
  4. kWh is energy, not power. A kilowatt-hour = 1000 W × 3600 s = 3.6 MJ. Bills charge energy; power is the rate.
  5. Average vs instantaneous. P = W/Δt averages over the interval; P = Fv is the rate right now (use the speed at that instant).
Common mistake: answering a “power” question with joules — giving the work instead of the rate. If the question says “how fast” or gives a time, divide by it.

Worked examples

Four problems, easiest first. Always ask: work known, or force-times-speed?

Example 1 — basic: W = 600 J in Δt = 20 s

  1. Divide. P = 600 / 20 = 30 W.
  2. Interpret. 30 joules per second — a dim light bulb's worth of power.
Common mistake: answering 600 J (“the power is 600”). Joules are work; the question asked for the rate. Divide by the time.
Your turn — W = 900 J in 30 s. P = ?

Answer: 30 W. 900/30 = 30 W.

Example 2 — lifting: m = 50 kg raised 2 m in 5 s

  1. Work first. W = mgh = 50 × 9.8 × 2 = 980 J.
  2. Divide by time. P = 980 / 5 = 196 W.
  3. Feel it. ~200 W is a strong human effort — plausible for hoisting 50 kg in 5 s.
Common mistake: P = mg/t (forgetting the height) — “power” without any distance. No displacement, no work, no power.
Your turn — m = 20 kg lifted 3 m in 4 s. P = ?

Answer: 147 W. W = 20×9.8×3 = 588 J; P = 588/4 = 147 W.

Example 3 — P = Fv: engine force 1500 N at 20 m/s

  1. Multiply. P = 1500 × 20 = 30,000 W = 30 kW.
  2. Why this form. No time given, none needed — force and speed carry the rate directly.
  3. Check. 30 kW ≈ 40 hp — a modest car cruising. Plausible. ✓
Common mistake: using P = Fv with the average speed when the question asks for power at top speed. P = Fv is instantaneous — use the speed at that moment.
Your turn — F = 800 N at v = 25 m/s. P = ?

Answer: 20,000 W (20 kW). 800 × 25 = 20,000 W.

Example 4 — the electric bill: 100 W bulb for 3 hours

  1. Energy = power × time. E = 100 W × 3 h = 300 Wh = 0.3 kWh.
  2. In joules. 0.3 kWh × 3.6 MJ/kWh = 1.08 MJ.
  3. Remember. The bill charges kWh (energy); the bulb's rating is W (power). Different quantities, different units.
Common mistake: “the bulb used 100 watts per hour” — watts per hour is meaningless. It's 100 watts for 3 hours = 0.3 kWh of energy.
Your turn — a 60 W bulb for 5 hours. Energy in kWh?

Answer: 0.3 kWh. 60 × 5 = 300 Wh = 0.3 kWh.

Memorization tips

  • Say it aloud: “power is work over time — the speed of the energy flow.”
  • Two doors: W and t known → P = W/t. F and v known → P = Fv. Pick the door with the given quantities.
  • Time in seconds for watts. Minutes and hours convert first — a 60× error is the classic.
  • kWh = energy. Power × time = energy. The bill measures what you used, not how fast.
  • Same work, less time, more power. The ratio shortcut: halve the time → double the power.
  • Human scale: ~100 W sustained (a cyclist), ~1000 W brief sprint. Use these to sanity-check answers.

Final challenge

Five mixed questions — rates, Fv, and the kWh trap. Score 5/5 and power is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

What's the difference between power and energy?

Energy (joules) is the total amount transferred; power (watts) is how fast it's transferred. A 60 W bulb for 1 hour uses 60 Wh of energy at a 60 W rate — amount vs speed.

When do I use P = Fv instead of P = W/t?

When you know the force and the speed but not the time or distance — engines, cyclists, rockets. It's the same formula with W = FΔx and Δx/Δt = v substituted.

Is a kilowatt-hour a unit of power?

No — it's energy: 1 kWh = 1000 W × 3600 s = 3.6×106 J. 'Kilowatt-hour' reads as power×time = energy. Power alone is just kilowatts.

Can power be negative?

Yes: P = F·v goes negative when force opposes velocity (braking, friction). Negative power means energy is being drained from the object — consistent with negative work.

What's average vs instantaneous power?

Pavg = W/Δt averages over an interval; P = Fv (or dW/dt) is the rate at one instant. A car's engine rating is closer to peak instantaneous; your electric bill uses average over time.

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