A rocket ascends from rest in Earth\'s gravitational field, by ejecting exhaust

Question

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 = ?ym where m is the instantaneous mass of the rocket and y 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 ( yu-g )/b. Calculate the time when the velocity is one-half of the terminal velocity.

Data: u = 26.1 m/s; b = 1.6 s?1.

Explanation / Answer

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



3. The attempt at a solution
I get dv=-udm-(g+bv)dt; dm=-?m
so dv=u?-(g+bv)dt

solving for v:
v(t)=(1/b)e^((-u?/b)t)-(g/b)=( yu-g )/b.

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