# Variable mass system

1. Sep 15, 2016

### erisedk

1. The problem statement, all variables and given/known data
A truck hauling a big tank of oil starts at position x = D (Chicago) and heads due west (–x direction) toward its destination at x = 0 (Des Moines). At Chicago, the total mass of the loaded truck is M and the mass of oil it is carrying is λM. (Thus M(1–λ) is the truck’s "tare" mass = the mass of the truck when it is empty.) The driver starts from rest at time t = 0 with his engines set to deliver a constant force of magnitude F throughout the trip.

Unfortunately, the trucker's oil tank is leaking: it is losing oil at a constant rate-per-unit-distance of
dm / dx = λ M / D . Here, m is the total mass of the truck and its load of oil. NOTE: the truck is losing oil, so
dm is negative, but dx is negative too since the truck is heading west; that's why dm / dx is a positive constant. Ignore the small change in m from the truck's consumption of gasoline (it's tiny compared to the truck's mass).

Calculate m(x) in terms of x and the given constants D, M, λ, and/or F.

2. Relevant equations
$m\frac{dv}{dt} = -\frac{dm}{dt}v^{ex} + F^{EXT}$

3. The attempt at a solution
I wish I could even start this problem, but all I see is that in this case $v^{ex}$ which is the exhaust velocity, i.e. the relative velocity of the oil with respect to truck, is zero.

That leaves me with $m\frac{dv}{dt} = F^{EXT}$
And I don't really know what to do with dv/dt at this point.