Posted by
**Jake** on
.

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