Question

In the arrangements shown in figure masses of each ball is $1kg$ and mass of trolley is $4kg$. In the figure shell of mass $1kg$ moving horizontally with velocity $v=6m/sec$ collides with the ball and get stuck to it then its maximum deflection (in degree) of the thread (length $1.5m$) with verticle is:

Answer 1

Find speed of the ball after collision.
Conserving the momentum of two balls before and after collision in horizontal direction

$1\times 6=(1+1)\times v$

$v=3m\sec$

Find speed of the trolley.
Given that trolley can move on frictionless surface at the time of maximum deflection.

Now at the instant of maximum deflection ball will be moving horizontal with same speed as of trolley (i.e,, velocity of ball w.r.t trolley will be 0),
Let it be $v_0$.

By conservation of momentum,
$2\times 3=(4+2)\times v_0$
$\Rightarrow v_0=1m/s$

Find deflection of the thread with vertical.
Now conserving energy.

$\frac{1}{2}(m_1+m_2)v^2$ = $\frac{1}{2}(M+m_1+m_2)v_0^2+mgl(1-\cos \theta )$

$\frac{1}{2}\times 2\times 3^2$ =

$\frac{1}{2}\times 6\times 1^2+2\times 10\times 1.5(1-\cos \theta )$

$\Rightarrow \cos \theta =0.8$

$\Rightarrow \theta ={37}^{\circ }$