**Fourth Equation**

**Relation between initial velocity, final velocity and displacement or position **

**v ^{2} = v_{0}^{2} + 2as or, v_{x}^{2} = v_{x0}^{2} + 2a (x – x_{0})**

Let an object move at a particular direction with uniform acceleration ‘a’. Let at t = 0 its initial velocity be v_{0} and at t = t, the final velocity ‘v’ and during this time interval the displacement of the object is ‘s’. Now, we know,

v = v_{0} + at … … … (1)

and s = v_{0}t + ½ at^{2} … … … (2)

Now squaring both sides of equation (1), we get;

v^{2} = (v_{0} + at)^{2 }= v_{0}^{2} + 2 v_{0}at + a^{2}t^{2}

= v_{0}^{2} + 2a (v_{0}t + ½ at^{2})

= v_{0}^{2} + 2as [using equation (2)] … … … (3)

For one dimensional notion, say along X-axis let v_{0} = v_{x0}, v = v_{x}, a = a_{x} and s = x – x_{0}, where x_{0} is the initial position at t = 0 and x the final position at t = t. Then from equation (3) we get**;**

**v _{x}^{2} = v_{x0}^{2} + 2a_{x} + 2a_{x} (x – x_{0})**