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Homework Help: Rocket with retarding force

  1. Jun 23, 2011 #1
    1. The problem statement, all variables and given/known data
    A rocket ascends from rest in Earth's gravitational field, by ejecting exhaust with constant speed u. Assume that the rate at which mass is expelled is given by dm/dt = −γm where m is the instantaneous mass of the rocket and γ is a constant; and that the rocket is retarded by air resistance with a force mbv where b is a constant.
    Determine the velocity of the rocket as a function of time. Here is a hint: The terminal velocity is ( γu−g )/b.

    Calculate the time when the velocity is one-half of the terminal velocity.
    Data: u = 31.9 m/s; b = 1.2 s−1.



    2. Relevant equations
    dp/dt=m(dv/dt)



    3. The attempt at a solution
    I got the diff eq down to:

    dv=-u(dm/m)-(g+bv)dt

    I'm not quite sure what I am doing wrong, I divide by -(g+bv)
    then solve from there to get -b*ln(g+bv)=uγt, but for some reason I dont think this is correct. Help, thanks.
     
  2. jcsd
  3. Jun 23, 2011 #2
    Do we take into account the fact that [itex]F_g = GMm/x^2[/itex] changes as the rocket goes higher, or are we assuming a constant [itex]F_g = mg[/itex]?
     
  4. Jun 23, 2011 #3
    assume Fg= -mg
     
  5. Jun 23, 2011 #4
    Alright. It's not working out for you because you have misapplied Newton's law: [itex]F = \frac{dp}{dt} = \frac{d}{dt}(mv) = m\frac{dv}{dt} + v\frac{dm}{dt}[/itex] by the product rule. Usually, m is constant, so dm/dt = 0, but in this case, dm/dt is given to be -γm.

    Also, what is the net force on the rocket?
     
  6. Jun 23, 2011 #5
    Fnet=-mg-mbv
    so dp/dt=F gives you :
    (-mg-mbv)dt=mdv+udm
    then,
    dv=uγ-(g+bv)dt

    that's right I just dont know how i keep messing up on solving the diff eq
     
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