- #1
Order
- 97
- 3
Homework Statement
Just have to ask one more question . I have two problems which I don't understand in Kleppner Kolenkow. They seem to contradict each other in my view. The first is 3.9 and the second is 3.10.
3.9 A freight car of mass M contains a mass of sand m. At t = 0 a constant horisontal 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 = 0.
3.10 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 the speed when the mass of sand, m has been transferred.
Homework Equations
F = dP/dt
The Attempt at a Solution
I try to find the differentials for these two problems.
3.9 P(t) = (M + m - dm/dt*t)v
P(t + dt) = (M + m - dm/dt*(t + dt))(v + dv) + dm/dt*dt*v
The last term dm/dt*dt*v is what I don't understand. The physical meaning of it being that the car gets an extra "boost" when mass flows out. The full differnatial equation becomes:
dP/dt = dv/dt*(M + m - dm/dt*t)
Lets look at the differentials of the other problem.
3.10 P(t) = Mv + vbt
P(t + dt) = M(v + dv) + (v + dv)b(t + dt)
In the same spirit as of the last problem I want to add a term b*dt*v, because the car is slowed down when sand at rest hits the car of speed v. The full differential equation becomes:
dP/dt = dv/dt*(M + bt) +bv
which is correct from check.
My question is, why add an extra term in 3.9 and not in 3.10?