How Is the Rocket's Mass Ratio Determined for Specific Speed Goals in Space?

[noppawit]

Consider a rocket that is in deep space and at rest relative to an inertial reference frame. The rocket's engine is to be fired for a certain interval. What must be the rocket's mass ratio (ratio of initial to final mass) over that interval if the rocket's original speed relative to the inertial frame is to be equal to (a) the exhaust speed (speed of the exhaust products relative to the rocket) and (b) 2.0 times the exhaust speed?
In this problem, it is about momentum. So I think I can use only P=mv equation.
I tried by let M=rocket's mass, V=rocket's velocity, m=exhaust's mass, v exhaust's velocity

For (a) I start: $$\sum$$P_i = $$\sum$$P_f
----------->>>>>> (m+M)(V_i) = MV_f-mv_f

For (b) I start: $$\sum$$P_i = $$\sum$$P_f
----------->>>>>> (m+M)(V_i) = MV_f-m*2v_f


Am I correct? If yes, I don't know how to solve this equation until I get $$\frac{m+M}{M}$$ = ......

Please help me.
Thank you very much.

 

[LowlyPion]

Consider a rocket that is in deep space and at rest relative to an inertial reference frame. The rocket's engine is to be fired for a certain interval. What must be the rocket's mass ratio (ratio of initial to final mass) over that interval if the rocket's original speed relative to the inertial frame is to be equal to (a) the exhaust speed (speed of the exhaust products relative to the rocket) and (b) 2.0 times the exhaust speed?



In this problem, it is about momentum. So I think I can use only P=mv equation.



I tried by let M=rocket's mass, V=rocket's velocity, m=exhaust's mass, v exhaust's velocity

For (a) I start: $$\sum$$P_i = $$\sum$$P_f
----------->>>>>> (m+M)(V_i) = MV_f-mv_f

For (b) I start: $$\sum$$P_i = $$\sum$$P_f
----------->>>>>> (m+M)(V_i) = MV_f-m*2v_f


Am I correct? If yes, I don't know how to solve this equation until I get $$\frac{m+M}{M}$$ = ......

Please help me.
Thank you very much.

You are right in observing that the problem is about momentum. Given that the rocket was at rest within the frame then what must the total momentum in the system be?