The 2 ends of train ,moving with a constant acceleration,pass a certain with velocity u and v .Show that velocity with which the middle point of train passes the same point is root of(usquare +v square)/2

Using the equation of kinematics for velocities, acceleration and displacement,

 

At the velocity at the mid-point of the train is v_m 

and that can be found using the same kinematical equation.

 

At the midpoint, the 'u' is zero and the displacement made is,

s_m , such that s_m = s/2.

 

Using equation (a) and (b)

Hence it is proved that the velocity of a train at its midpoint is the square root of half of the difference of the squares of their final and initial velocity.