Question

A system of two blocks $A$ and $B$ are connected by inextensible massless strings as shown. The pulley is massless and frictionless. A bullet of mass $m$ moving with a velocity $u$ as shown hits the block $B$ and gets embedded into it. Find the impulse imparted by tension force to the system

Answer 1

Let velocity of $B$ and $A$ after collision has magnitude $v$
At the time of collision tension $=T$
Impulse provided by tension $=\int Tdt$
For bullet
$-\int Tdt=mv-mu...\left(1\right)$
For block $B$
$\int (F-T)dt=mv....\left(2\right)$
Consider change of momentum of $(B+$ bullet)
$mu-\int Tdt=2mv....\left(i\right)$
Consider change of momentum $A$
$\int Tdt=3mv....\left(ii\right)$
from (i) and (ii)
$mu=5mv$
$v=u/5$
Hence, impulse $\int Tdt=3mv=3m\left(\frac{u}{5}\right)=\frac{3mu}{5}$