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**A momentum question - having conceptual difficulties - please help..!!**

## 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

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 = o.

## Homework Equations

F = d(P)/dt

## The Attempt at a Solution

I applied the above equation to the system of "freight car".

Which is,

F = d(P)/dt, where P is the momentum of the freight car at any instant of time.

Now P = mv, where m and v are the total mass and velocity of the freight car at any instant. Now, d(P)/dt should be

v(dm/dt) + m(dv/dt) .................. (right?)

But I looked up the solution, and over there it is instead given,

d(P)/dt = d(mv)/dt = m(t) x (dv/dt)

where m(t) is the mass of the freight car at time 't'.

So essentially, in the solution, m has been taken out of differentiation. My question is, why is it so?

What is wrong with my differential equation, d(P)/dt = d(mv)/dt = v(dm/dt) + m(dv/dt) ?

Please help......

Thanks