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 $k$.

Maximum compression of the spring is $\sqrt{\frac{ymgh}{k}}$. Find $2y$.

Answer 1

Initial linear momentum of the system is zero and there is no external force on the system, hence at the time of maximum compression of spring the linear momentum of the system should be zero. Hence final kinetic energy should be zero.
$\Delta U+\Delta K=0\Rightarrow \Delta K=K_f-K_i=0$
Hence, $\Delta U_{gravity}+\Delta U_{spring}=0$
$-mgh+\frac{1}{2}k{x}^2=0$

$x=\sqrt{\frac{2mgh}{k}}=\sqrt{\frac{ymgh}{k}}$

$y=2$

$2y=4$