A goods train accelerating uniformly on a straight railway track, approaches an electric pole standing on the side of track. Its engine passes the pole with velocity $$u$$ and the guard’s room passes with velocity $$v.$$ The middle wagon of the train passes the pole with a velocity.

Solution :
Let $$'S'$$ be the distance between two ends $$'a'$$ be the constant acceleration.
As we know $${v^2} - {u^2} = 2aS\,\,{\text{or,}}\,\,aS = \frac{{{v^2} - {u^2}}}{2}.$$
Let $$v$$ be velocity at mid point.
Therefore, $$v_c^2 - {u^2} = 2a\frac{S}{2} \Rightarrow v_c^2 = {u^2} + aS$$
$$v_c^2 = {u^2} + \frac{{{v^2} - {u^2}}}{2} \Rightarrow {v_c} = \sqrt {\frac{{{u^2} + {v^2}}}{2}} $$