Find velocity of rocket that obtains more gas from gas cloud

[Scronin267]

Homework Statement

To solve this problem you will need to construct a differential equation. A picture of the situation will help. Ignore all gravitational forces. A Rocket ship of structural mass M and fuel mass m, begins at rest relative to a gas cloud. The ship burns fuel at the rate ω which is exhausted with speed v relative to the ship. The ship obtains pure fuel from the gas cloud at the rate 2ω.
Find the velocity of the ship as a function of time.

Homework Equations

The Attempt at a Solution

All I can think of is using momentum... P initial = 0, but I do not now how to construct the equation for P final.

 

[gneill]

You need to make a better attempt. Perhaps do some research on the rocket equation and its derivation first so that you can at least prepare a credible list of relevant equations.

 

[Scronin267]

I am having a hard time describing this situation, and getting proper cancelations\simplifications.
To start we need to find an equation describing the momentum... I think. This would look like.
P = P_ship + P_exhausted fuel + P_gas cloud = 0
P_ship = (M_total)(v+dv)
P_exhausted = (v_ship - v_fuel)(dm_exhausted)
P_gas cloud = (v_ship)dm_intake
we also know that the fuel m = m_o+2ω-ω=m_o+ω
Using this information I am unable to get a differential equation that has an meaning.

 

[haruspex]

I am having a hard time describing this situation, and getting proper cancelations\simplifications.
To start we need to find an equation describing the momentum... I think. This would look like.
P = P_ship + P_exhausted fuel + P_gas cloud = 0
P_ship = (M_total)(v+dv)
P_exhausted = (v_ship - v_fuel)(dm_exhausted)
P_gas cloud = (v_ship)dm_intake
we also know that the fuel m = m_o+2ω-ω=m_o+ω
Using this information I am unable to get a differential equation that has an meaning.

You need to be more systematic in the analysis, using differentials everywhere that is appropriate.
In time dt, let the ship change speed from u to u+du.
What mass of fuel did it burn in that time? What was the exhaust speed of that fuel relative to the gas cloud? What momentum did it take with it?
What mass of new fuel did it take on?
What is the new momentum of the ship plus its load of fuel? What was the momentum change in time dt?
What equation can you now write?