# Speed of moving mass as it loses mass

## Homework Statement

A Freight Car of Mass M contains a mass of sand m. At t=0, a constant horizontal force i 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 a constant rate dm/dt. Find the speed of the freight care when all the sand is gone. Assume the car is at rest at t=0

## Homework Equations

F= ma
mv = mv - conservation of momentum?

## The Attempt at a Solution

If there were no sand leaking from the bottom of the train, then the train would be accelerating at a constant acceleration because F= (M+m)a and the masses would be constant. However, the mass is changing, so the acceleration will be increasing over time. I'm just not sure how I can put these into a mathematical answer

## The Attempt at a Solution

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tiny-tim
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Welcome to PF!

Hi lei123! Welcome to PF! ## Homework Statement

A Freight Car of Mass M contains a mass of sand m. At t=0, a constant horizontal force i 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 a constant rate dm/dt. Find the speed of the freight care when all the sand is gone. Assume the car is at rest at t=0

If there were no sand leaking from the bottom of the train, then the train would be accelerating at a constant acceleration because F= (M+m)a and the masses would be constant. However, the mass is changing, so the acceleration will be increasing over time. I'm just not sure how I can put these into a mathematical answer
Just do it step by step …

what is the mass at time t? …

what is the acceleration at time t? ehild
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1.

## Homework Equations

F= ma
mv = mv - conservation of momentum?

The momentum is not conserved as there is an external force, and also the mass is changing. Moreover, F=ma is valid for a body of constant mass. Newton formulated his second law in a more general form, so it was valid even for changing mass:

"The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed."

F*dt = d (mv)

that is F = d(mv)/dt = v*dm/dt +m*dv/dt.

ehild