Physics - Center of Mass Question with Solution | TestHub

The block's kinetic energy at the bottom of the slope is $mgh$.

This energy is dissipated by friction as the block slides across the carriage.

The work done by friction is $\mu mg\ell$. The carriage also accelerates.

By conservation of momentum,

$mv=(M+m)V$,

where $v$ is the block's velocity before impact and $V$ is the combined velocity. The friction force causes the block to decelerate and the carriage to accelerate until they move together. Equating the initial kinetic energy to the work done by friction plus the final kinetic energy:

$mgh=\mu mg\ell (1+\frac{m}{M})$.

Therefore, $h=\mu (1+\frac{m}{M})\ell$.