Question

A block of mass $m$ is pushed with a velocity ${\nu }_0$ along the surface of a trolley car of mass $M$. if the horizontal ground is smooth and the coefficient of kinetic friction between the block and plank is $\mu$, find the minimum distance of relative sliding between the block and plank.

Answer 1

Decide system: Block and plank
Observation: No external force acting on system in horizontal direction
Conclusion: Linear momentum of the system will be conserved.
As friction in kinetic nature and system.
When relative sliding between block and trolley stops, velocity (V).
Problem solving: Using conservation of linear momentum in horizontal direction.
$m{\nu }_0+0=(m+M)V⟹V=\frac{m{\nu }_0}{(m+M)}$ (i)
Now using WE theorem from ground frame.
$W_{ext}+W_{int}=\delta K$
$W_{ext}=0$
$W_{int}=W_{friction}=-fx=-\mu mgx$
$0=(-\mu mgx)=\left(\frac{1}{2}\right(m+M)V^2-\frac{1}{2}m{\nu }_0^2)$ (ii)
From (i) and (ii), relative distance of sliding
$x=\frac{{\nu }_0^2}{2(1+\frac{m}{M})\mu g}$