# Mass Ratio Q: Solving for Final Speed of 8km/s

• cycam
In summary, the equation states that the exhaust speed, v_ex, is relative to the rocket, and the final mass, m_f, is relative to the initial mass, m_i. The mass ratio, MR, is found by solving for v_delta, which is the difference in speed between the exhaust and initial speeds.f

## Homework Statement

A single-stage rocket is fired from rest from a deep-space platform, where gravity is negligible.

If the rocket burns its fuel in a time of 50.0 s and the relative speed of the exhaust gas is v_ex=2100 m/s, what must the mass ratio m_{0}/m be for a final speed v of 8.00 km/s (about equal to the orbital speed of an Earth satellite)?

## Homework Equations

v-v0= -v_exln(m/m0) = v_exln(m0/m)

## The Attempt at a Solution

having a rough time understanding the equation, any tips or hints?

Any help would be greatly appreciated.

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I wonder if you are supposed to make a formula for the mass ratio as a function of time (it does vary). Is the mass lost at a constant rate?

It doesn't say that it has to be which i would assume it shouldn't be. All I know about the question is what you see. So to answer your question, i don't know. I was hoping someone would be able to answer that lol.

Does the rocket start from rest? If so, and you only care about the mass ratio and t=50 sec., then you just have to work out the log equation.

Yes, it appears that it is starting from rest. How would I go about finding the mass ratio? Also, the log equation?

How do I go about finding the mass of the rocket? (is that even the right question to ask)

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updated the original question.

I'll help with the equation. It is known as the Tsiolkovsky rocket equation.

$$\Delta v = v_{\text{exhaust}} \ln\left(\frac{m_{\text{init}}}{m_{\text{final}}}\right)$$

where

$v_{\text{exhaust}}$ is the exhaust speed relative to the rocket
$m_{\text{init}}$ is the initial (pre-burn) mass of the rocket
$m_{\text{final}}$ is the final (post-burn) mass of the rocket
$\ln(x)$ is the natural logarithm function.

The problem gives $v_{\text{exhaust}}$ and $\Delta v$ and simply asks for the mass ratio ${m_{\text{init}}}/{m_{\text{final}}}$. Can you proceed with this?

Let me do the math really quickly and i'll show you what I got.

so is it MR=v_delta/v_exln?

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ok, i got it. thank you =)