Consider an object moving with **uniform acceleration**, a. Let u be the **initial velocity** (at time t = 0), v, **velocity after time t** and S, **displacement** during this time interval. There are certain relationships between these quantities.

Acceleration = Change in velocity / Time interval

a = (v - u)/t

**v = u + at**

This is called as the **first equation of motion**.

Displacement = (average velocity) × (time interval)

s = (v + u)/2 × t

s = (u + at + u)/2 × t

**s = ut + ½at ^{2}**

This is called the **second equation of motion**.

If object starts from rest, u = 0

s = ½at^{2}

Thus, the displacement of an object undergoing a constant acceleration is proportional to t^{2}, while the displacement of an object with constant velocity (zero acceleration) is proportional to t.

Now,

a = (v - u)/t and s = (v + u)/2 × t

Multiply them

a.s = (v^{2} - u^{2})/2

**v ^{2} = u^{2} + 2as**

This is called as **third equation of motion**. In case of motion under gravity a can be replaced by g.