Final Velocity of Multistage Rocket | Homework Equations

[GB89]

Homework Statement

I need to find the final velocity of a multistage rocket (accelerating from rest in free space) that burns its fuel in two stages. In the first stage, it burns 3/10 of its initial mass as fuel. Then it jettisons its first-stage fuel tank, which has a mass of 1/10 the initial mass of the rocket. Then, it burns 3/10 of its initial mass as fuel.

Homework Equations

I think that the equation is:

Final Velocity = V(exhaust) times ln(m0/m) + V(exhaust) times ln(m0/m).

The Attempt at a Solution

Is the m0 in the second part of the equation 6/10 initial mass, or 9/10 initial mass (does the fuel burned in the first stage get subtracted too)?

I think the answer is V(exhaust) times ln(1/.7) + V(exhaust) times ln(.6/.3), giving a final answer of 1.0498 times the exhaust velocity. Is this correct?

EDIT: PLEASE DISREGARD. I just noticed that this question was answered in a "Similar Thread." My appologies.

 

[Jhero]

Thank you for your question! You are correct in your approach to finding the final velocity of the multistage rocket. However, there is one small error in your equation. The m0 in the second part of the equation should be 6/10 initial mass, as the fuel burned in the first stage does not get subtracted. This means that the correct equation is:

Final Velocity = V(exhaust) times ln(1/.7) + V(exhaust) times ln(.6/.6)

Using this equation, you will get a final answer of 0.6931 times the exhaust velocity.

I hope this helps! Keep up the good work in your studies of rocket propulsion.