Question

A block of mass m and a pan of equal mass are connected by a string which is going over a smooth light pulley as shown in the figure. The system is at rest initially when a particle of mass ‘m’ falls on the pan and sticks to it. If the particle strikes the pan with a speed v, find the speed with which the system moves just after the collision.

• A. $\frac{v}{2}$
• B. $\frac{v}{3}$
• C. $\frac{v}{6}$
• D. $\frac{2v}{3}$

Answer 1

The correct option is B

$\frac{v}{3}$

Let the speed after collision be V.
As there is a sudden change in the speed of the block, the tension must change by a large amount during the collision.

Let N= magnitude of the contact force between the particle and the pan
T = tension in the string
dt = The time duration of the collision
Consider the impulse imparted to the particle. The force is N in the upward direction and the impulse is Ndt. This is equal to the change in its momentum.
Thus,
Ndt=mv-mV (i)
Similarly for the pan: (N-T)dt=mV (ii)
And for the block: Tdt=mV (iii)
Adding (ii) and (iii),
Ndt=2mV
Comparing with (i),
mv-mV = 2mV
Therefore, $V=\frac{v}{3}$