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.

 

[nlightner]

Hello,

Thank you for sharing the problem with me. I understand your approach to the problem and it seems like you have the right idea. However, there are a few things that need to be clarified and corrected in your solution.

Firstly, the equation F = M(t)(dv/dt) is not correct. The correct equation should be F = (M + m)(dv/dt), where m is the mass of the sand that has been transferred into the car at time t. This is because the total mass of the car at any given time is not just M, but also includes the mass of the sand that has been added.

Secondly, when you integrate both sides of the equation dv = F(dt)/(M + bt), you should not include the constant of integration. This is because the initial condition is given as the car starting from rest, so the constant of integration would be 0.

Finally, the correct solution would be v = (F/b)(ln(M+m) - ln(M)). This can be simplified to v = (F/b)ln((M+m)/M).

I hope this helps clarify the solution for you. If you have any further questions, please let me know. Good luck!