The Kinetic Energy Theorem

Statement of the theorem

The kinetic energy theorem relates the change in an object’s kinetic energy between two instants to the sum of the work done by the forces acting on it:

delta Ec = Ec_final - Ec_initial = sum of W(forces)

In other words, if the forces applied perform positive work, the kinetic energy increases (the object accelerates); if the work is negative, the kinetic energy decreases (the object decelerates).

Work done by a force

The work done by a constant force F over a distance d, in the same direction, is:

W = F * d (in joules)

If the force opposes the motion (such as friction or braking), the work it does is negative.

Example

A car weighing 1,000 kg is travelling at 20 m/s, then brakes and comes to a halt. The change in its kinetic energy is:

delta Ec = 0 - 1/2 * 1,000 * 20² = -200,000 J

The work done by the braking forces is therefore -200,000 J: this is the energy that had to be dissipated (mainly as heat) to bring the vehicle to a halt.

Common pitfall

Do not confuse delta Ec (the change) with Ec itself. delta Ec can be negative, whereas kinetic energy Ec is always positive or zero; it is never negative.