**Third Equation of Dimensional Motion**

In one dimensional motion a body moves along a straight line. So quantities associated with motion, for example displacement, velocity, acceleration etc., have only one component (moving along X-axis will have X-component and Y and Z components will be zero). In deriving equations of linear motion we will consider that the body is moving along X-axis. In that case subscripts associated with different quantifies of motion may be omitted. Normally, v_{x} will be represented by v and a_{x} by a.

**Third Equation**

*Equation of motion relating position or displacement, acceleration and time: S =v _{0}t + ½ st^{2}.*

**or, x = x _{0} + v_{x0}t + ½ a_{x}t^{2}**

If a body is moving with uniform acceleration its average velocity is equal to half of the sum of the initial velocity v_{0} and the final velocity v.

Thus, average velocity v = (v_{0} + v) / 2

But we know, **v = v _{0} + at**

So, average velocity, v = (**v _{0} + v_{0} + at) / 2**

**= v _{0} + ½ at **… … … (1)

Again we know average velocity,

**v = s/t = displacement/time**

**or, s = vt **… … … (2)

Now putting v from equation (1) to equation (2); we get:

**S = (v _{0} + ½ at) t = v_{0}t + ½ at^{2}** … … … (3)

For one dimensional motion, say along X-axis, let the initial position at t = 0 is x_{0} and at time t = t. the position is x. So, the change of position, displacement, s = x – x_{0}. Now, if initial velocity is v_{x0} and the final velocity v and the acceleration a, then from equation (3) we get;

S = x – x_{0} = v_{x0}t + ½ a_{x}t^{2}

or, **x = x _{0} + v_{x0}t + ½ a_{x}t^{2}**