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Work and the Work–Energy Theorem
Classical Mechanics
In everyday language, work means effort. In physics it has a precise meaning: work is done when a force moves an object through a distance. The work-energy theorem turns that definition into a powerful problem-solving tool, linking the work done on an object directly to how its speed changes. Once you can use it, many motion problems become simple energy bookkeeping.
What work means in physics
Work is defined as force times the distance moved in the direction of the force: **W = F d** when the force is constant and acts along the motion. Its unit is the joule (J), and 1 J = 1 N·m. Only the component of force along the displacement does work: pushing straight down on a trolley that moves sideways does no work, and carrying a bag horizontally at a steady height does no work against gravity. If the force opposes the motion, as friction does, the work is negative because it removes energy from the object.
The work-energy theorem
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: **W_net = ΔKE = ½ m v² − ½ m u²**. If the net work is positive, the object speeds up; if it is negative, it slows down; if it is zero, the speed stays the same. This is useful because it lets you find a final speed without tracking the acceleration at every moment. It also explains braking: the brakes do negative work on a car, removing its kinetic energy until it stops, which is why stopping distance grows with the square of the speed.
Kinetic and potential energy
Kinetic energy, **KE = ½ m v²**, is the energy of motion — doubling the speed quadruples the kinetic energy. Gravitational potential energy, **PE = m g h**, is the energy stored by lifting an object to a height h above a reference level. When an object falls, potential energy is converted into kinetic energy, and the total mechanical energy stays constant if friction is negligible. A roller coaster trades height for speed and back again in exactly this way. Work done against gravity increases potential energy, which is why lifting a heavy box slowly still costs real effort.
📝 Worked example: pushing a box
A box of mass 2 kg is pushed from rest with a constant net force of 10 N over a distance of 5 m. Work done = F d = 10 N x 5 m = 50 J. By the work-energy theorem, this equals the gain in kinetic energy: KE = 50 J. So 1/2 x 2 kg x v² = 50 J, which gives v² = 50. v = √50 = 7.07 m/s. Answer: the box reaches a speed of about 7.1 m/s.
Common mistakes
- Counting work when there is no displacement. Correction: holding a heavy bag still does no physics work, however tiring it feels, because the distance moved is zero.
- Using the full force when it acts at an angle. Correction: only the component of force along the displacement does work.
- Thinking kinetic energy doubles when speed doubles. Correction: KE depends on v², so doubling the speed quadruples the kinetic energy.
Practice questions
- How much work is done by a 20 N force pushing a box 3 m in the direction of the force?
Show answer
W = F d = 20 x 3 = 60 J.
- A 4 kg object moving at 5 m/s is brought to rest. How much work was done on it?
Show answer
KE before = 1/2 x 4 x 25 = 50 J. KE after = 0, so net work = -50 J.
- What is the gravitational potential energy of a 5 kg book lifted 2 m? Use g = 9.81 m/s².
Show answer
PE = m g h = 5 x 9.81 x 2 = 98.1 J.
Frequently asked questions
Can work be negative?
Yes. Work is negative when the force acts opposite to the displacement, as friction does — it removes kinetic energy.
What is a joule?
A joule is the SI unit of work and energy: 1 J is the work done by a 1 N force moving an object 1 m.