Derive the equation for Velocity - displacement relation.

Explanation

Velocity of - displacement relation
The acceleration is given by the first derivative of velocity with respect to time.
`a=(dv)/(dt)=(dv)/(ds) (ds)/(dt)=(dv)/(ds)v`
[since ds/dt=v] where s is displacement traversed.
This is rewritten as `a=(1)/(2) (dv^(2))/(ds) (or) ds=(1)/(2a) d(v^(2))`
Integrating the above equation, using the fact when the velocity changes from `u_(2)` to `v_(2)`, displacement changes from 0 to s, we get
`int_(0)^(s)ds=int_(u)^(v)(1)/(2a)d(v^(2))`
`therefore s=(1)/(2a)(v^(2)-u^(2))`
`therefore v^(2)=u^(2)+2as`
We can also derive the displacement s in terms of initial velocity u and final velocity v.
From the equation `v=u+at` we can write, `at = v-u`
Substitute this in equation `s= ut+(1)/(2)at^(2)`, we get
`s=ut+(1)/(2)(v-u), s=((u+v)t)/(2)`