Hi. This one has been bugging me for a while. We briefly covered this in lectures getting just the main points with no derivations and I would like to fully understand this. I have a couple of questions and any help would be greatly appreciated.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

The main jist is this: A rocket is initially at rest, with respect to inertial frame K. Then the engines are fired and the rocket starts moving. Assume the exhaust gases are ejected at a constant rate v' in the opposite direction with respect to the rocket. Let m1 be the rest mass of the rocket that will, of course, change when material is being ejected.

The main goal is to calculate the velocity as a function of remaining mass m1. There are a couple of steps beforehand though.

Firstly, I note that the system rocket plus exhaust is isolated, thus total energy and momentum are conserved. Considering one infinitesimal step of rocket acceleration to the next, I let dm2 be the mass of a small chunk of exhausted material ejected within each infinitesimal step, v1 be the velocity of the rocket and v2 the velocity of the exhaust gas, as seen in K. Thus m1, v1 and v2 vary while dm2 is ejected.

2. Relevant equations

Energy conservation Law (given): [tex]d\left(\frac{m_{1}c^2}{\sqrt{1-\frac{v_{1}^2}{c^2}}}\right) + \frac{c^2 dm_{2}}{\sqrt{1-\frac{v_{2}^2}{c^2}}} = 0[/tex]

Momentum Conservation (given): [tex]d\left(\frac{m_{1}v_{1}}{\sqrt{1-\frac{v_{1}^2}{c^2}}}\right) + \frac{v_{2} dm_{2}}{\sqrt{1-\frac{v_{2}^2}{c^2}}} = 0[/tex]

Before I go any further, I am wondering why, in both conservation equations, we consider the differential of the entire first term, but only dm2 in the second term?

Secondly, I'm probably being really dense here, and I apologise for that, but if I wanted to calculate dm1 and dm2 as functions of m1, v1, dv1 and v2, how can this be done? It's clearly just a mathematical manipulation of the above expressions, but those differentials are putting me off - I'm probably just missing a trick.

Any help with this would be greatly appreciated, and should allow me to get a bit deeper into the problem.

Thanks in advance

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# Homework Help: Relativistic Rocket Science

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