# Derive the third equation of motion: v2 - u2 = 2as

For getting the third law of motion equation, we use both of the equations of motion

S = ut + 1/2at2__________(1)

v = u + at _____________ (2)

t = (v – u)/a

Substitute the value of t in equation (1)

S = ut + 1/2at2

$$S = u\frac{(v-u)}{a}+\frac{1}{2}a(\frac{v-u}{a})^{2}$$

2aS = 2u(v − u) + (v − u)2

2aS = 2uv − 2u2 + v2 − 2uv + u2

2aS = v2 − u2

v2 = u2 + 2aS