Question

A block of mass $m$ is released from rest from a height $h$ onto a smooth sledge of mass $M$ fitted with an ideal spring of stiffness $k$.

If the velocity of the block and sledge just before the block touches the spring is $\sqrt{\frac{a{m}^{2}gh}{M(M+m)}}$. Find the value of $a$.

Answer 1

As no external force in horizontal direction, the linear momentum will be conserved in horizontal direction. Let the velocity of block and sledge be $v$ and $V$ respectively up to the case when the block reaches on horizontal surface on the sledge.
$mv=MV$ (i)
Mechanical energy is also conserved in this case
$\Delta K+\Delta U=0$
$(\frac{1}{2}m{v}^2+\frac{1}{2}M{V}^2-0)+(-mgh)=0$ (ii)
$\frac{1}{2}m{v}^2+\frac{1}{2}M{\left(\frac{m}{M}v\right)}^2=mgh$
$v=\sqrt{\frac{2Mgh}{M+m}}$ and $V=\sqrt{\frac{2m^{2}gh}{M(M+m)}}$