An open top railroad car (initially empty and of mass M.) rolls with negigible friction along a straight horizontal track and passes under the sprout of a sand conveyor. When the car is under the conveyor, sand is dispensed from the conveyer in a narrow stream at a steady rate delta-M/delta-t=C and falss vertically from an average height h above the florr of the railroad car. The car has an initial speed v. and sand is filling it from time t = 0 to t = T. Express your answers to the following in terms of the given quatities and g.
a) Determine the mass M of the car plus the sand that it catches as a function of time t for 0 < t < T.
b) Determine the speed v of the car as a function of time t for for 0 < t < T.
c) i.) Determine the initial kinetic energy of the empty car.
ii.) Determine the final kinetic energy K final of the car and its load
iii.) Is kinetic energy conserved? Explain.
d.) Determine the expressions for the normal force exerted on the car by the tracks at the following times
i.) before t=0
ii.) for for 0 < t < T.
iii.) after t=T
Conservation of Momentum: m1(v1i) +m2(v2i) = m1(v1f) + m2(v2f)
The Attempt at a Solution
I got part A by using Calculus: CT + Mo
But I have a question about part b. I was going through the homework archives on this website and I found a previous poster said that:
v(t) = momentum / the mass as a function of time:
v(t) = p/m(t) = m0v0/(m0 + Ct).
My question is:
Why does v(t) = momentum / the mass as a function of time?
And is the reason why can you use m0v0 (the initial momentum) as the momentum for V(t) for the entire time from o<t<T because momentum is conserved throughout the entire time interval?