Rocket ship conservation of mometum

[dawozel]

Homework Statement

Interstellar Spaceship
An interstellar spaceship with initial mass Mo is at rest at the edge of a small, spherical nebula (gas cloud). At t= 0, the engines begin to fire, ejecting gas out the back at constant speed u relative to the rocket. The mass of the rocket decreases at a constant rate: dm/dt= -β, where β is a positive constant. As the spaceship accelerates along a diameter of the nebula, the only external force it experiences is a frictional resistance
proportional to its velocity: f = -kv,where k is a positive constant.

a) use change in momentum to find an expression for dv/dt
in the nebula as a function of time t, velocity v, and constants.
DO NOT JUST STATE A RESULT!

b) While the spaceship is in the nebula, find, in terms of time t
and constants: i. An expression for v(t), the speed of the spaceship.
ii. An expression for x(t), the distance the rocket has traveled.
NOTE Use the initial conditions to remove undetermined
constants.

Homework Equations

none I guess, probably use conservation of momentum

The Attempt at a Solution

so we started with $$dm/dt = -β$$
you then turn it into a separable differential equation and you get
$$m= -βt +Mo$$

we also know
$$f = -kv = mdv/dt$$

now i plugged in for mass

$$f = -kv = mdv/dt = (-βt +Mo)dv/dt$$

and finally you get
$$dv/dt= -kv/(-βt +Mo)$$

now this may be right but it does not use conservation of momentum to solve, I was hoping you could help me here

 

[dauto]

Seems like you forgot to include the engine thrust.