Question

A rocket ascends from rest in a uniform gravitational field by ejecting exhaust with constant speed u.
Assume that the rate at which mass is expelled is given by dm/dt=mk, where m is the instantaneous mass of the rocket and k is a cosntant, and that the rocket is retarded by air resistance with a force bv where b is a constant. find the velocity of the rocket as a function of time.

I have worked it down to the last step but am having trouble finishing the integration to find v(t).

F_net = F_thrust - F_grav - F_air.resist
ma = (dp/dt) - mg - bv
(dv/dt) = (uk - g) - (bv/m)

m = m_o*e^kt ---> not sure if this part is right??

so (dv/dt) = (uk - g) - b*m_o*e^-kt*v

v = dx/dt so multiply by dt to get the integration equation of:

dv = (uk-g)dt - b*m_o*e^-kt*dx

my attempt so far:

v(t) = (uk-g)t - ????

***I am stuck on integrating the:
b*m_o*e^-kt*dx

Answer 1

I think that the answer comes out to be

v(t)=(uk-g)t - b*m_o*(e^-kt)*x(t)

Answer 2

because b and m_o are constants and i believe that e^-kt is taken as a constand as well. Leaving dx to be integrated between 0 and x(t).

am i correct? more opinions welcomed please