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Variable mass system

  1. Sep 15, 2016 #1
    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.
    Please help.
  2. jcsd
  3. Sep 15, 2016 #2


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    Staff: Mentor

    I don't see where velocity or force comes into play at all. The question states that the rate of oil loss is constant per unit distance. Thus the total amount of oil lost depends only on the distance traveled and should be a simple linear function of position.
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