In the figure shown, the pulley is smooth and the string is inextensible. An impulse $J$ is applied

Question

In the figure shown, the pulley is smooth and the string is inextensible. An impulse $J$ is applied to the block $m$, then the net impulse acting on the system ( $m+2 m$ ) is

Professor PC Thomas Classes · Answer

Let $T$ be the impulse generated in the string, then by using Impulse-Momentum theorem.
$
m v-0=J-T
$
and $\quad(2 m) v-0=T$
On adding, (1) and (2), we get
$
3 m v=J \quad 	ext { or } \quad v=\frac{J}{3 m}
$
and $\quad T=2 m v=\frac{2}{3} J$
∴ the net impulse acting on the system is
$
J_{n e t}=J-T^{\prime}=J-\frac{2 J}{3}=\frac{J}{3}
$