- #1
Mr Davis 97
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Homework Statement
A sand-spraying locomotive sprays sand horizontally into a freight car. The locomotive and the freight car are not attached. The engineer in the locomotive maintains his speed so that the distance to the freight car is constant. The sand is transferred at a rate ##\displaystyle \frac{dm}{dt}##, with a velocity ##\vec{u}## relative to the locomotive. The freight car starts from rest with an initial mass ##M_0##. Find the speed of the freight car for all time t.
Homework Equations
Change in momentum
The Attempt at a Solution
Since this is a flow of mass problem, we will use the concept of mass transfer and momentum, rather than F = ma.
First, we isolate the system such that we initially have a stationary freight car and some sand traveling towards the car.
Taking this system as it is, we can find the change of momentum.
##P(t) = \Delta m u##, where ##u## is the relative velocity of the sand with respect to the freight car, and ##\Delta m## is the mass of the little portion of sand we are analyzing.
##P(t + \Delta t) = M \Delta v + \Delta m \Delta v##
##P(t + \Delta t) - P(t) = M \Delta v + \Delta m \Delta v -\Delta m u ##
##\displaystyle \frac{\Delta P}{\Delta t} = M \frac{\Delta v}{\Delta t} + \frac{\Delta m \Delta v}{\Delta t}- \frac{\Delta m}{\Delta t} u##
##\displaystyle \frac{dP}{dt} = M \frac{dv}{dt} - \frac{dm}{dt} u##
Is this correct so far? If so, how do I proceed? Is ##\displaystyle \frac{dP}{dt} = 0##?