Solving Mass Flow Problem Homework

[Mr Davis 97]

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 $\displaystyle \frac{dP}{dt} = 0$?

 

[Mr Davis 97]

Bump. Some help would be nice...

 

[gneill]

I might approach the problem in a slightly different way. Let $r = \frac{dm}{dt}$ be the rate at which sand is delivered to the freight car. The relative velocity between the arriving sand and the freight car is fixed at $u$ thanks to the matching speed of the locomotive.

What can you say about the mass of the freight car + sand as a function of time? Can you write an expression for it in terms of $M_o , r, \text{ and } t$?

What is the rate of delivery of momentum to the freight car (that's $\frac{dp}{dt}$)? Can you write an expression for it in terms of $r, u, \text{ and }t$?

What's another name for $\frac{dp}{dt}$?