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Simple Rocket Equation in Gravity

  1. Dec 16, 2014 #1
    1. The problem statement, all variables and given/known data
    I'm having trouble deriving the equation for the velocity of a rocket in constant G given that it has constant exhaust velocity

    2. Relevant equations

    I know that a=dv/dt=U/M(dM/dt)-g

    3. The attempt at a solution
    so from here dv=U/M dM-gdt

    v=U ∫1/M dM - g ∫dt v from 0 to v, M from Minitial to Mfinal, and t from 0 to t

    which leads to v=-ln[Minitial/Mfinal]-gt

    HOWEVER, everywhere I look, the answer is positive ln[Minitial/Mfinal] -gt and I can't seem to get why this is so
  2. jcsd
  3. Dec 16, 2014 #2


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    Homework Helper

    From conservation of momentum you get the equation MΔv-UΔm=-Mg, where Δm is the exhausted mass during Δt time. But it is negative of the change of mass of the rocket. So the differential equation for the rocket mass is M dv/dt+UdM/dt=-Mg ---->## v= -\int _{Minitial}^{Mfinal}(\frac{dM}{M})##
  4. Dec 16, 2014 #3
    Ah, i see what I was doing wrong, my answer is correct if U is taken to to be negative
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