Work-Energy Theorem

Statement

The work done by a net force on a particle is equal to the change in its kinetic energy.

Mathematical Form

$$ W = \Delta K $$

Kinetic Energy

$$ K = \frac{1}{2}mv^2 $$

Derivation

From Newton’s second law:

$$ F = ma $$

Using:

$$ a = \frac{dv}{dt}, \quad v = \frac{dx}{dt} $$

We write:

$$ a = v \frac{dv}{dx} $$

So:

$$ F = m v \frac{dv}{dx} $$

Multiply both sides by $dx$:

$$ F dx = m v dv $$

Integrate:

$$ \int F dx = \int m v dv $$

$$ W = \frac{1}{2}mv^2 - \frac{1}{2}mu^2 $$

Thus:

$$ W = \Delta K $$