# Rocket with retarding force

## Homework Statement

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.

dp/dt=m(dv/dt)

## 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.

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Do we take into account the fact that $F_g = GMm/x^2$ changes as the rocket goes higher, or are we assuming a constant $F_g = mg$?

assume Fg= -mg

Alright. It's not working out for you because you have misapplied Newton's law: $F = \frac{dp}{dt} = \frac{d}{dt}(mv) = m\frac{dv}{dt} + v\frac{dm}{dt}$ 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?

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