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Rocket accelerating thru a dust cloud

  1. Dec 25, 2006 #1
    hi...
    a rocket of mass M is flying through a dust cloud the cloud has a density of P. the rocket's cross section is A. every dusticle the rocket colides with becomes permenantly attached.
    the rocket is ejecting material (as a propellant) at the same rate that it assimilates it. the speed of the ejection relative to the rocket is
    [tex]V_g [/tex]
    find the rockets acceleration as a function of its speed V


    2. Relevant equations
    [tex]
    F = M{{dv} \over {dt}} + (u - v){{dm} \over {dt}}

    [/tex]


    3. The attempt at a solution
    i'v found dm, i think:
    [tex]
    dm = APdx = APVdt

    [/tex]

    so presumably, to find
    the acceleration i did this:
    [tex]
    \eqalign
    & dV = adt - V_0 = V_1 - V_0 \cr
    [/tex]
    [tex] & - V_0 dm = - V_g dm + MV_1 \cr[/tex]
    but now im totally stuck...
    any help? please?
     
    Last edited: Dec 25, 2006
  2. jcsd
  3. Dec 25, 2006 #2

    OlderDan

    User Avatar
    Science Advisor
    Homework Helper

    This does not look right

    [tex]
    dV = adt - V_0 = V_1 - V_0 \cr
    [/tex]

    The definition of acceleration is just a = dV/dt

    Since the mass of the rocket is not changing, you can treat the collision//expulsion of a mass dm using conservation of momentum. It does not matter that the gas expelled is not the same piece of matter as the dm that is captured. Think of a head-on collision between two objects of mass M and dm with M moving at an initial velocity V and dm at rest. The final velocity of dm is known in terms of V and the relative exhaust velocity. Solve for the final velocity of M and find the rate of change of velocity dV/dt. This will look a lot like your force equation, but I don't think you have that equation quite right. Maybe it's just a matter of interpretation. What force are you representing by that equation?
     
  4. Dec 27, 2006 #3
    ok.. i think you are right. so using conservation of momentum i can approach it like this:
    in an infinitesimally small ammonut of time, the following equation holds true:

    [tex] VM = (V + dV)M + (V - V_g )dm [/tex]
    [tex]VM = (V + dV)M + (V - V_g )APdx [/tex]
    [tex] VM = VM + Mdv + (V - V_g )APdx [/tex]
    [tex]{{(V - V_g )APdx} \over {dt}} = M{{dv} \over {dt}} [/tex]
    [tex] {{(V - V_g )APV} \over M} = a [/tex]

    p.s. how do i make a newline in tex?? \newline doesnt seem to work
     
    Last edited: Dec 27, 2006
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