Physics › 🚀 Mechanics

Work, Energy & Power

Push a box, drop a ball, or climb a flight of stairs — it's all the same currency changing hands.

High schoolAP Physics 1 level
💡
The big idea: Energy is a single quantity that shows up in different disguises — motion, height, heat — and it never appears or vanishes on its own; it only transfers or transforms. Work is the name we give to that transfer when a force pushes something through a distance, and power is simply how fast the transfer happens.
🎯 By the end, you'll be able to
  • Calculate the work done by a constant force acting over a distance, including when the force isn't fully aligned with the motion.
  • Compute kinetic energy from mass and speed, and explain why doubling speed quadruples it.
  • Apply conservation of mechanical energy to find an unknown speed or height in a frictionless system.
  • Calculate power as the rate of energy transfer and connect it to everyday devices.
📎 You should already know
  • Newton's Laws of Motion
  • Vectors and scalars
  • Basic algebra (solving for one variable)

One quantity, many costumes

Here's the trick that makes energy so useful: it's just one thing wearing different outfits. A stretched bow, a boulder at the top of a cliff, a speeding car, a hot cup of coffee — all of these are energy, just stored in different forms. Physics gets remarkably simple once you stop asking "where did the energy come from?" and start asking "what form was it in before, and what form is it in now?"

Work, kinetic energy, and potential energy are just the vocabulary for tracking that bookkeeping. Once you can speak the language, you can predict speeds, heights, and forces without ever touching Newton's second law directly.

🔑 The core principle: energy is conserved

In a system where the only forces doing work are conservative ones (like gravity), the total mechanical energy — kinetic plus potential — stays exactly the same. It just keeps trading between the two forms. A falling apple isn't gaining energy from nowhere; it's cashing in potential energy for kinetic energy, dollar for dollar.

\[ W = F d \cos\theta \]
Work equals force times displacement times the cosine of the angle between them. If the force pushes straight along the direction of motion, \(\theta = 0\) and this simplifies to \(W = Fd\). If the force is perpendicular to the motion, it does zero work — no matter how strong it is.
\[ KE = \tfrac{1}{2} m v^2 \]
Kinetic energy grows with the square of speed. Double your speed and you don't double your kinetic energy — you quadruple it. That's why highway crashes are so much more destructive than parking-lot fender benders.
\[ PE_{grav} = mgh \]
Gravitational potential energy depends only on mass, gravitational acceleration \(g \approx 9.8\ \text{m/s}^2\), and height above some reference level you choose.
🎮 Interactive: watch energy trade forms LIVE
Drag the object to different heights or speeds and watch kinetic and potential energy swap back and forth while their total stays flat. Try releasing it from rest and see how the two bars cross exactly halfway down.

Power: the same energy, but how fast?

Two people can lift the same box to the same shelf — same work, same energy transferred. But if one does it in two seconds and the other takes a lazy ten minutes, something is clearly different between them. That "something" is power: the rate at which energy is delivered or converted.

Power is why a small electric motor can still lift a heavy elevator (given enough time), while a sprinter's legs generate enormous power for a very short burst. It's not about how much energy you can produce — it's about how fast you can push it through.

\[ P = \dfrac{W}{t} \]
Power is work (or energy transferred) divided by the time it took. Measured in watts, where \(1\ \text{W} = 1\ \text{J/s}\). For a force moving something at constant velocity, this is equivalent to \(P = Fv\).
📝 Worked example: A 60 kg skier starts from rest at the top of a frictionless slope 20 m high. How fast is she moving at the bottom?
  1. Choose the bottom of the slope as the reference height, so potential energy there is zero.
  2. Find the potential energy at the top: \(PE = mgh = 60 \times 9.8 \times 20 = 11{,}760\ \text{J}\).
  3. Since the slope is frictionless, mechanical energy is conserved: all of that potential energy converts into kinetic energy at the bottom, so \(KE = 11{,}760\ \text{J}\).
  4. Solve \(\tfrac{1}{2}mv^2 = 11{,}760\) for \(v\): \(v^2 = \dfrac{2 \times 11{,}760}{60} = 392\), so \(v = \sqrt{392} \approx 19.8\ \text{m/s}\).
  5. Notice the mass canceled out along the way — in the absence of friction or air resistance, a heavier skier and a lighter skier released from the same height arrive at the same speed.
✓ About 19.8 m/s (roughly 71 km/h) at the bottom of the slope.
📝 Worked example: A 50 kg hiker climbs a 4 m flight of stairs in 5 seconds. What is her average power output?
  1. Find the work done against gravity: \(W = mgh = 50 \times 9.8 \times 4 = 1{,}960\ \text{J}\).
  2. Divide by the time taken: \(P = \dfrac{W}{t} = \dfrac{1{,}960}{5} = 392\ \text{W}\).
✓ About 392 W — roughly a third of the power a typical microwave draws, just to climb some stairs.
⚠️ Where the simple version breaks down

Conservation of mechanical energy only holds when friction and air resistance are negligible — that's why textbook slopes are always suspiciously "frictionless." In the real world, some mechanical energy continuously leaks away as heat and sound. The energy itself is never destroyed (that's the broader law of conservation of energy), but it stops being available as clean, usable kinetic or potential energy. So if a real skier's speed at the bottom comes in a bit lower than your calculation predicts, friction — not a broken formula — is almost always the reason.

Check your understanding

1. A satellite moves in a perfect circle around Earth, with gravity providing the centripetal force. How much work does gravity do on it per orbit?
Work requires a component of force along the direction of motion. Since the gravitational force always points toward the center while the velocity is always tangent to the circle, the angle between them is 90 degrees the entire time, so \(\cos\theta = 0\) and the work done is zero.
2. A 2 kg ball is moving at 3 m/s. What is its kinetic energy?
\(KE = \tfrac{1}{2}mv^2 = \tfrac{1}{2}(2)(3)^2 = \tfrac{1}{2}(2)(9) = 9\ \text{J}\).
3. A 1,500 kg car speeds up from 10 m/s to 20 m/s. How much does its kinetic energy increase?
Initial KE = \(\tfrac{1}{2}(1500)(10)^2 = 75{,}000\ \text{J}\). Final KE = \(\tfrac{1}{2}(1500)(20)^2 = 300{,}000\ \text{J}\). The increase is \(300{,}000 - 75{,}000 = 225{,}000\ \text{J} = 225\ \text{kJ}\) — notice it's far more than double, since kinetic energy scales with speed squared, not speed.
4. Two balls, one twice as heavy as the other, are dropped from the same height with no air resistance. How do their speeds compare just before hitting the ground?
Setting \(mgh = \tfrac{1}{2}mv^2\) and solving for \(v\) gives \(v = \sqrt{2gh}\), which has no mass term at all — it cancels out. Without air resistance, gravity accelerates every mass equally, so both balls reach the same speed.
✅ Key takeaways
  • Work (W = Fd cos θ) is the transfer of energy that happens when a force acts through a distance — a force perpendicular to motion does no work at all.
  • Kinetic energy (½mv²) scales with the square of speed, so small speed increases cause much bigger energy increases.
  • In a frictionless system, mechanical energy (kinetic + potential) is conserved — energy just trades forms, letting you find unknown speeds or heights without touching forces directly.
  • Power (P = W/t) measures how fast energy is transferred, not how much total energy is involved.