You and a friend both carry a 20-kg bag up the same stairs. You take 10 seconds; your friend takes 60. The work you did is identical (same mass, same height). But something’s different — your power was 6× higher. Power is the rate at which work is done — and it’s what separates a family car’s engine from a Ferrari’s.
The core idea in one paragraph 📌
Power ($P$) = rate of doing work or transferring energy. Formula: $P = W/t$, or equivalently $P = \vec{F} \cdot \vec{v}$ (instantaneous power). SI unit: watt = joule per second. The industrial energy unit: kilowatt-hour (kWh) = 3.6 MJ = power × time (not the other way around!). The traditional but still-common unit: horsepower (hp) = 746 W. Efficiency is the ratio of useful output power to input power.
Three equivalent formulas 📐
$$\boxed{P = \frac{W}{t} = \frac{\Delta E}{t}}$$
Average power — total work divided by total time.
$$\boxed{P = F \cdot v \cos\theta = \vec{F} \cdot \vec{v}}$$
Instantaneous power — the fully-physical form, force times velocity. Proof: $P = \tfrac{dW}{dt} = \tfrac{d(F d)}{dt} = F \tfrac{dd}{dt} = Fv$ (for a force along the motion).
Watts, kilowatts, kWh — these are different things! ⚡
The most-confused topic in everyday physics:
| Symbol | Meaning | Base unit |
|---|---|---|
| W (watt) | Power | J/s |
| kW (kilowatt) | Power × 1000 | J/s |
| Wh (watt-hour) | Energy = power × time | 3600 J |
| kWh (kilowatt-hour) | Energy = power × time | 3.6 × 10⁶ J |
Your electricity bill is in kWh, not kW — because you pay for “amount of energy consumed,” not instantaneous power.
Example: a 2-kW heater running for 3 hours:
$$E = P \times t = 2\ \text{kW} \times 3\ \text{h} = 6\ \text{kWh} = 6 \times 3.6\ \text{MJ} = 21.6\ \text{MJ}$$
Power scale — familiar things 📊
| Source | Power |
|---|---|
| Human thinking (brain) | ~20 W |
| Human basal metabolic rate | ~100 W |
| Amateur cyclist | ~200 W |
| Tour de France champion (peak) | ~400 W |
| LED light bulb | 10 W |
| Laptop | 30–60 W |
| Microwave oven | 1000 W |
| Family car engine | ~100 kW (~130 hp) |
| Tesla Model S Plaid motor | 760 kW |
| Large wind turbine | 3 MW |
| Mid-size nuclear reactor | 1 GW |
| Total electricity use of Earth (avg) | ~20 TW |
| The Sun (total output) | $3.8 \times 10^{26}$ W |
Efficiency 🎯
$$\boxed{\eta = \frac{P_\text{useful}}{P_\text{input}} \times 100\%}$$
No real machine is 100% efficient — the second law of thermodynamics forbids it. Typical values:
| System | Efficiency |
|---|---|
| Gasoline car engine | ~25% |
| Diesel engine | ~40% |
| Electric-car motor | ~95% |
| Thermal power plant | ~35% |
| Commercial solar panel | ~22% |
| Plant photosynthesis | ~1–2% 🌱 |
| Human body (muscle work) | ~25% |
| LED bulb | ~40% (rest is heat) |
| Incandescent bulb (old) | ~5% (95% heat!) 💀 |
Instantaneous power — car example 🚗
A 1200-kg car accelerates at $2$ m/s². When its speed hits $20$ m/s:
$$F = ma = 1200 \times 2 = 2400\ \text{N}$$
$$P = Fv = 2400 \times 20 = 48{,}000\ \text{W} = 48\ \text{kW} \approx 64\ \text{hp}$$
Power grows with speed — which is why acceleration gets slower at higher speeds (a fixed-power engine can’t push a fast car as hard).
Python analysis 🐍
1) Stair-climbing power — comparing people
m, g = 70, 9.8 # 70-kg person
h_stairs = 3.5 # climb 3.5 m
def stair_power(time_s):
W = m * g * h_stairs
return W / time_s
for label, t in [("Walking", 15), ("Normal", 8), ("Running", 3.5), ("Elite", 2)]:
P = stair_power(t)
print(f"{label:10s} ({t}s) → power = {P:.0f} W ≈ {P/746:.2f} hp")
# Walking ~160W, running ~688W, elite ~1200W (>1.6 hp!)
2) Monthly electricity bill
# A few devices with daily usage hours
devices = [
("Refrigerator (avg)", 150, 24),
("LED lights x 5", 50, 5),
("Air conditioner", 1500, 6),
("Washing machine", 2000, 0.5),
("Laptop", 50, 8),
("Microwave", 1000, 0.3),
]
daily_kWh = 0
for name, P_W, hours in devices:
E = P_W * hours / 1000
daily_kWh += E
print(f"{name:22s} {P_W:>5d}W × {hours:>4.1f}h = {E:.2f} kWh/day")
monthly_kWh = daily_kWh * 30
cost = monthly_kWh * 0.15 # assume $0.15/kWh
print(f"\nMonthly consumption: {monthly_kWh:.1f} kWh")
print(f"Approximate cost: ${cost:.2f}")
3) Efficiency and wasted energy
def efficiency_analysis(P_input, eta):
"""Compute useful and wasted power."""
P_useful = P_input * eta
P_wasted = P_input * (1 - eta)
return P_useful, P_wasted
systems = [
("Gasoline engine", 80_000, 0.25),
("Diesel engine", 80_000, 0.40),
("EV motor", 80_000, 0.95),
("Incandescent bulb", 100, 0.05),
("LED bulb", 100, 0.40),
]
for name, P_in, eta in systems:
P_useful, P_wasted = efficiency_analysis(P_in, eta)
print(f"{name:20s} η={eta*100:4.1f}% → useful {P_useful:>7.0f}W, waste {P_wasted:>7.0f}W")
Take-home summary 🎁
$P = W/t = Fv$ — two complementary forms. Unit is watt (not kWh!). Your bill is in kWh because you’re paying for energy, not power. The scale from a 20-W brain to a $10^{26}$-W Sun spans 25 orders of magnitude. Efficiency is useful/input — from 5% for an incandescent bulb to 95% for an EV motor. Python calculates stair-climbing power, monthly bills, and efficiency losses cleanly 😎.
Congratulations — Chapter 3 is complete 🎉
“Nice to know” box: Horsepower — a marketing trick 🐴
James Watt built an improved steam engine in 1769. Problem: people didn’t know what “engines” were — they knew horses. His brilliant solution: he measured how much work an average horse could do per second (~150 lbs, 220 ft, per minute = 550 ft·lb/s ≈ 746 W). Then he said “my engine has the power of 10 horses.” People instantly understood. A 250-year-old marketing trick that stuck. The SI unit “watt” is named for him and is the official one, but you’ll still see horsepower on every car brochure — not because it’s more precise, but because it’s more intuitive to human brains.
Test yourself 📝
References and further exploration 📚
Articles and reference
- Wikipedia: Power (physics), Horsepower, Kilowatt hour, Energy conversion efficiency
- HyperPhysics — Power
- Feynman Lectures — Vol. I, Ch. 13: Work and Potential Energy — power section
Videos (YouTube)
- Veritasium: Why we still use horsepower
- MIT OCW 8.01 Walter Lewin: Power, work rate
- Real Engineering: The physics of Tour de France cyclists
- CGP Grey: What is a kilowatt-hour?
External simulators
On this site 🔗
- نسخهی فارسی — توان
- Previous section — Work and Internal Energy
- Related — Conservation of Mechanical Energy
🎉 Congratulations — Chapter 3 (Work, Energy, and Power) is complete. You now have kinetic energy, work, the work-energy theorem, potential energy (gravitational and elastic), conservation of mechanical energy, conversion to internal energy, and power. Ready for Chapter 4 — Temperature and Heat 🌡️🚀?
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