Question

A trolley of mass 300 kg carrying a sand bag of 25 kg is moving uniformly with a speed of $27km/h$ on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of $0.05kg{{s}^{-1}}$. What is the speed of the trolley after the entire sand bag is empty?

Answer 1

Hint: To solve this question first find if there are any external forces acting on the system. If the external force is zero then we can use the law of conservation of momentum to find the final speed of the system. So, find the final mass of the system, as the initial mass and speed is given, we can find the final speed.

Complete step by step answer:
Initially the mass of the trolley is 300 kg and the mass of the sandbag on the trolley is 25 kg. the system is moving on a frictionless track with velocity $27km/h$.

Since the system is moving with constant velocity and there is no frictional force, the net force on the system is zero.

Sand is leaking out of the hole in the trolley at the rate $0.05kg{{s}^{-1}}$.
Finally, after all the sand was leaked, the mass of the system will be only 300 kg.

Since, there is no external force acting on the system we can say that the momentum will be conserved for the system.

So, we can write,

$\begin{align}
  & {{M}_{i}}{{v}_{i}}={{M}_{f}}{{v}_{f}} \\
 & \left( 300kg+25kg \right)27km/h=\left( 300kg \right){{v}_{f}} \\
 & 325kg\times 27km/h=300kg\times {{v}_{f}} \\
 & {{v}_{f}}=\dfrac{325kg}{300kg}27km/h \\
 & {{v}_{f}}=29.25km/h \\
\end{align}$

So, by using the law of conservation of momentum we can say that the velocity of the system after the sandbag is emptied on the trolley will be $29.25km/h$
The final velocity will be $29.25km/h$

Note: Since we are asked about the velocity of the sand at the point where the bag of sand is empty, the final velocity does not depend on the rate of falling of sand. So, we do not use this part to solve the question.