Question

A man of mass $m$ moves on a plank of mass $M$ with a constant velocity $u$ with respect to the plank. as shown in Fig.
a) If the plank rests on a smooth horizontal surface, determine the velocity of the plank.
b) If the man travels a distance $L$ with respect to the plank, find the distance travelled by the plank with respect to the ground.

Answer 1

a) let ${\nu }_1$ and ${\nu }_2$ be the velocities of the man and the plank w.r.t ground. Then we have
$[{\overrightarrow{\nu }}_{man}{]}_{plank}=[{\overrightarrow{\nu }}_{man}{]}_{ground}-[{\overrightarrow{\nu }}_{man}{]}_{ground}$
The, $[{\overrightarrow{\nu }}_{man}{]}_{plank}=[{\overrightarrow{\nu }}_{man}{]}_{ground}+[{\overrightarrow{\nu }}_{man}{]}_{ground}$
Or ${\nu }_1=u-{\nu }_2$
Initially, the system is at rest. Therefore,
$0=m{\nu }_1-M{\nu }_2$
or $0=m(u-{\nu }_2)-M{\nu }_2$
or ${\nu }_2=\frac{mu}{m+M}$
b) Let $x$ be the displacement of plank in backward direction, the displacement of mass is $L-x$ in forward direction.
$\Delta c_{CM}=\frac{M(-x)+m(L-x)}{M+m}$
$⟹0=Mx+m(L-x)⟹x=\frac{mL}{M+m}$