Result

Work-energy theorem

W_{net} = Δ K = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{i}^{2}

Statement

The net work done on an object equals its change in kinetic energy.

Symbols and units

When it applies

Any object with constant mass, from the net (total) force acting on it.

Don't use when

Relates net work to kinetic energy only — the work of a single force among several need not equal the change in kinetic energy.

Derivation

Start from Newton's second law and the definition of work along a straight line:

W_{net} = \int F d x = \int ma d x

Use a = v\,dv/dx (the chain rule applied to acceleration) to change the variable of integration:

W_{net} = \int m v d v = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{i}^{2}

Worked examples

Speed from net work

A 2.0 kg object starts at rest; 40 J of net work is done on it:

v_{f} = \frac{2 W _{net}}{m} = \frac{2 ( 40 )}{2.0} = 6.3 m/s

Watch out for

This is the <i>net</i> work — from every force combined. The work of a single force need not equal the change in kinetic energy.

Connects to