The kinetic energy theorem

Reminder: kinetic energy

The kinetic energy of an object of mass m moving at speed v is:

Ec = 1/2 * m * v^2

It is expressed in joules (J), like work.

Statement of the theorem

The kinetic energy theorem relates the work of forces to the change in speed of an object: between an initial instant and a final instant, the change in kinetic energy equals the sum of the work done by all applied forces.

Ec(final) - Ec(initial) = sum of W(forces)

Concrete example

A car of mass m = 1000 kg travels at v1 = 20 m/s. The driver brakes and the combined friction forces perform resisting work W = -200000 J until it comes to a complete stop (v2 = 0).

Ec(initial) = 1/2 * 1000 * 20^2 = 200000 J Ec(final) = 0 J

Check: Ec(final) - Ec(initial) = -200000 J, which indeed matches the sum of the resisting work that slowed the car down.

Usefulness of the theorem

This theorem allows calculating a final speed without studying the motion in detail, using only the work of the forces. It is widely used to study braking, falls, or loops.

Classic pitfall

The theorem uses the sum of ALL forces (weight, reaction, friction, driving force...), not just a single isolated force. Forgetting a force is the most common mistake.