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Differential equation for a rocket

  1. Nov 9, 2014 #1
    1. The problem statement, all variables and given/known data
    A rocket through interstellar dust, no gravity. Solve for v with respect to mass.

    gif.latex?%5Cdot%7Bm%7D%3D%5Cfrac%7Bdm%7D%7Bdt%7D%3Dk.gif
    k is a constant.
    Drag = -bv

    2. Relevant equations

    gif.latex?m%5Cfrac%7Bdv%7D%7Bdt%7D%3D-%5Cdot%7Bm%7Du-bv.gif

    3. The attempt at a solution

    To start:

    %7Bdv%7D%7Bdm%7D%3D-ku-bv%20%5Crightarrow%20m%5Cfrac%7Bdv%7D%7Bdm%7D%3D-u-%5Cfrac%7Bb%7D%7Bk%7Dv.gif

    It should become a separable differential equation, but I am having a lot of trouble solving it for v(m). A hint that was given, is in the final equation you should have has m/m0 to a power.
     
  2. jcsd
  3. Nov 9, 2014 #2
    You could divide by ## -u - \frac{b}{k} v## and also by ##m## and then multiply by ##dm## to get

    $$\frac{dv}{-u - \frac{b}{k} v} = \frac{dm}{m}$$

    This is solvable, see if it gets you the answer you need--I feel like both terms' logarithmic solutions would cancel but because of the coefficients perhaps you will get the correct dependence when you solve for it.
     
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