Speed of car with changing mass problem

[bodensee9]

Hello:

An empty freight car of mass M starts from rest under an applied force F. At the same time, sand begins to run into the car at steady rate b from a hopper at rest along the track. Find speed when mass of sand m has been transferred.

I have attached a drawing.

So would this be: Let M(t) = mass as a function of time. So we have
M(t) = M + bt
And we have F = M(t)(dv/dt)
So F(dt) = M(t)dv
or dv = F(dt)/(M + bt)
So wouldn't we have
v = F/b*ln(M + bt)?
But somehow this is wrong?
Thanks.

 

[bucher]

You've got it right up to dv = F(dt)/(M + bt)

By doing a substitution of "M + bt" and then integrating accordingly, you should get

v = (F/b)*ln((M + bt)/M)

I think all that you did wrong was that you forgot to solve for the "t = 0" end of the integral.