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Question

A cart of mass M0 is moving with velocity v0. At t=0, water starts pouring into the cart from a container above the cart at the rate of λ kg/sec. Find the velocity of the cart as a function of time.


A
M0v0M0λt
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B
M0v0M0+λt
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C
M0v0M0+2λt
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D
None of these
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Solution

The correct option is B M0v0M0+λt
Mass of water poured per second = rate of change of mass of cart =dmdt=λ
Let the velocity of cart be v at any time t. Its initial velocity is v0.

The thrust force on the cart is given by
F=(dmdt)(urel)
where urel=v is the velocity of cart relative to the container in the horizontal direction.
F=λv

Now the net mass of the cart at any time t is (M0+λt) kg
Applying Newton's second law for cart in horizontal direction, considering thrust force (F) on it, as the net external force:
F=mdvdt
λv=(M0+λt)dvdt
Integrating both sides,
t0λdt(M0+λt)=vv0dvv
λλln(M0+λt)t0=lnv|vv0
[ln(M0+λt)lnM0]=lnvv0
ln(M0+λtM0)=ln(vv0)
Or, ln(M0M0+λt)=ln(vv0)
M0M0+λt=vv0
v=M0v0M0+λt

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