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Question

A cart loaded with sand moves along a horizontal floor due to constant force F=10 N acting in the direction of the cart's velocity vector. In the process, sand spills through a hole at the bottom at a constant rate, μ=5 kg/s . Find the velocity of the cart at the time t=5 sec, if at the initial moment t=0, the cart loaded with sand has mass m0=100 kg and its velocity was equal to zero. Friction is to be neglected. [Take ln(43)=0.288 and g=10 m/s2]

A
1.428 m/s
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B
0.823 m/s
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C
1.1 m/s
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D
0.576 m/s
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Solution

The correct option is D 0.576 m/s
Given,
Initial mass of the cart + sand m0=100 kg
Rate at which sand spills from the hole μ=5 kg/s

In this problem, the sand spills through a hole in the bottom of the cart. Hence, the relative velocity of the sand vr will be zero because it will have the same horizontal velocity as the cart at the moment of spilling.

Thus, Ft=0[as Ft=vr(dmdt)]
and the net force will be F only.
i.e Fnet=F
or m(dvdt)=F ...(i)
But here, m=m0μt
(m0μt)dvdt=F


Integating both sides,
v0dv=t0F(m0μt)dt
v=Fμ[ln(m0μt)]t0
v=Fμln(m0m0μt)
At t=5 s,
v=105ln(1001005×5)=2ln(4/3)
v=0.576 m/s

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