Unit 3 — Work, Energy, and Power
An energy-based way to analyze motion: kinetic energy, work, potential energy, conservation of energy, and power. Each section ends with a short practice set — click "Show answer" to check yourself.
Translational Kinetic Energy
Kinetic energy is the energy an object has because of its motion. The SI unit of all energy is the joule: 1 J = 1 kg·m²/s².
K = ½ · m · v²
- Kinetic energy is a scalar. It is never negative and has no direction.
- K depends on v²: doubling the speed quadruples the kinetic energy.
- K depends on the reference frame, because velocity does.
- The kinetic energy of a system is the sum of the kinetic energies of its parts.
1. A 2.0 kg cart moves at 3.0 m/s. What is its kinetic energy? What if its speed doubles?
Show answer
2. A 4.0 kg object has 50 J of kinetic energy. How fast is it moving?
Show answer
Work
Work is the transfer of energy to or from a system by a force acting through a displacement. Only the component of force along the displacement does work.
W = F · d · cos θ (θ = angle between F and displacement) W_net = ΔK = K_f − K_i (work–energy theorem)
| Angle θ | Work | Effect on the system |
|---|---|---|
| 0° (force along motion) | Positive | Energy added (speeds up) |
| 90° (force perpendicular) | Zero | No energy change |
| 180° (force opposes motion) | Negative | Energy removed (slows down) |
- Work is a scalar measured in joules.
- The normal force on a sliding object and the centripetal force in uniform circular motion do zero work, since they are perpendicular to the motion.
- On a graph of force vs. position, the area under the curve equals the work done.
- Kinetic friction does negative work and transfers energy into thermal energy.
Fnet first.1. A 10 N force acts at 60° above the horizontal on a crate that moves 5.0 m horizontally. How much work does the force do?
Show answer
2. A 2.0 kg object starts from rest and net work of 36 J is done on it. What is its final speed?
Show answer
Potential Energy
Potential energy is stored energy associated with the arrangement of objects in a system that interact through a conservative force (gravity or a spring). It belongs to the system, not to a single object.
| Type | Equation | Notes |
|---|---|---|
| Gravitational (near Earth) | ΔUg = m·g·Δh | Only the change in height matters; choose any zero level |
| Elastic (spring) | Us = ½·k·x² | x = stretch or compression from equilibrium |
| Gravitational (general) | Ug = −G·m₁·m₂ / r | Zero at infinite separation, so always negative |
- A force is conservative if the work it does depends only on start and end positions, not on the path. Then
W = −ΔU. - Spring potential energy is the same whether the spring is stretched or compressed by the same amount, since
xis squared. - The area under an Fs vs. x graph for a spring equals the stored elastic energy.
1. A 3.0 kg book is lifted 4.0 m at constant speed. By how much does the gravitational potential energy of the book–Earth system change? (g = 9.8 m/s²)
Show answer
2. A spring (k = 100 N/m) is compressed 0.20 m. How much energy is stored?
Show answer
Conservation of Energy
Energy cannot be created or destroyed, only transferred between objects or transformed between forms. For any system:
ΔE_system = W_external Isolated system (no external work, no friction): K_i + U_i = K_f + U_f With friction: K_i + U_i − f·d = K_f + U_f
- Choose the system first. Gravity and springs are internal forces when Earth or the spring is part of the system.
- Work done by nonconservative forces (friction, applied pushes) changes the total mechanical energy
K + U. - Friction converts mechanical energy into thermal (internal) energy:
ΔEth = fk·d. Total energy is still conserved. - Energy methods ignore the path taken, which makes them powerful for curved tracks, pendulums, and roller coasters.
Steps: pick initial and final points, choose a zero level for U, list K and U at each point, add any work by nonconservative forces, and solve.
1. A ball is dropped from rest at a height of 20 m. Ignoring air resistance, what is its speed just before it hits the ground? (g = 9.8 m/s²)
Show answer
2. A 2.0 kg block slides from rest down a ramp, dropping 5.0 m in height. It reaches the bottom at 6.0 m/s. How much energy was converted to thermal energy by friction? (g = 9.8 m/s²)
Show answer
Power
Power is the rate at which energy is transferred, or equivalently the rate at which work is done. Its SI unit is the watt: 1 W = 1 J/s.
P_avg = W / Δt = ΔE / Δt P = F · v · cos θ (instantaneous power for a force on a moving object)
- Two machines can do the same work, but the one with greater power does it in less time.
- Power is a scalar.
- When a force acts along the direction of motion,
P = Fv. - Another common unit: 1 horsepower ≈ 746 W. A kilowatt-hour (kW·h) is a unit of energy (3.6 × 10⁶ J), not power.
1. A student lifts a 50 kg box 3.0 m at constant speed in 5.0 s. What is the average power? (g = 9.8 m/s²)
Show answer
2. A car's engine exerts a forward force of 2000 N while the car moves at a constant 20 m/s. What power does the force deliver?