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Question

A cart filled with sand of mass m0=50 kg is moving with constant velocity because of a constant force F=10 N acting in the direction of the cart's velocity vector. Sand starts spilling through a hole in the bottom with a constant velocity at the rate of μ=10 kg/s.. Find the velocity of the cart at the time t=4 s. Friction is to be neglected. [Take ln(5)= 1.61]

A
2.7 m/s
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B
1.61 m/s
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C
3.2 m/s
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D
16.1 m/s
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Solution

The correct option is B 1.61 m/s

For a variable mass system with constant rate of ejection dmdt,
Fextudmdt=mdvdt
Fext=mdvdt+udmdt

Here, relative velocity of ejection u=0 (since sand has same horizontal velocity component as the cart)

Initially at t=0,m=m0
After time t, m=m0μt

F=mdvdt
m(dv)=Fdt

Integrating both sides and putting the value of m
v0dv=t0Fm0μtdt
v=Fμ[ln(m0m0μt)]
At t=4 s,
v=1010[ln(505010×4)]=1.61 m/s

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