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A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/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 0.05 kg $$s^{-1}$$. What is the speed of the trolley after the entire sand bag is empty ?


Solution

Since the hole is on the floor , that means sand is falling vertically with respect to trolley. Therefore there is no force in horizontal direction hence in horizontal direction momentum is conserved.
Let $$M=$$ mass of trolley
     $$m=$$ mass of sandbag
     $$v_1=$$ initial velocity 
     $$v_2=$$ final velocity ( to be found)  
Then $$P_1= (M+m)v_1$$ when the sand bag is empty the momentum is 
$$P_2= (M+0) v_2$$
Momentum is conserved in horizontal direction so
$$P_1= P_2$$
$$\Rightarrow = v_2= \dfrac{(M+m)}{M} v_1= \dfrac{300+25}{300} \times 27 \times \dfrac{5}{18} = 8.215 m/s$$

Physics
NCERT
Standard XI

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