- #1
Saitama
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Homework Statement
A freight car of mass M contains a mass of sand ##m##. At ##t=0## a constant horizontal force F is applied in the direction of rolling and at the same time a port in the bottom is opened to let the sand flow out at the constant rate dm/dt. Find the speed of the freight car when all the sand is gone. Assume the freight car is at rest at ##t=0##.
Homework Equations
The Attempt at a Solution
Let ##\frac{dm}{dt}=\lambda##.
At t=0, momentum of the system, P(0)=0.
At some later time t, ##P(t)=(M+m-\lambda t)v## where v is the velocity of car at time t.
When all the sand falls out, ##m=\lambda t \Rightarrow t=m/\lambda##. Hence ##P(m/\lambda)=Mv##.
$$F=\frac{P(t)-P(0)}{t}=\frac{P(m/\lambda)}{m/\lambda}=\frac{M \lambda v}{m}$$
Therefore, the speed of car is
$$v=\frac{Fm}{M\lambda}$$
Does this look correct?
Any help is appreciated. Thanks!