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I've looked at multiple sources on the internet and just can't wrap my mind around it.

If I reference the wikipedia page on it; http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

I am getting stuck on certain places.

$V_e=V-v_e $ Why does this occur?

What integration limits do we use to derive this and why?

$\Delta V\ = v_e \ln \frac {m_0} {m_1} $ (I am only concerned with this equation up to this part)

If someone could explain it from the beginning in an easier fashion that would be great but solving these doubts for me would also help.

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$V_e=V-v_e $ Why does this occur?

This is just saying that the exhaust velocity is measured with respect to the engine. If the rocket is moving forward ($V$), then the observed exhaust velocity ($V_e$) with respect to the ground (or other specified frame) is reduced by the engine's velocity.

And since we are concerned with the changes in velocity with respect to that frame, the change in exhaust velocity with respect to it has to be taken into account.

Does this page help at all? http://exploration.grc.nasa.gov/education/rocket/rktpow.html

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The article is a little wordy relative to what you're asking, so I'll offer a summary which encompasses the definition of the problem being solved. The equation of motion at work is a version of conservation of momentum.

$$ F = m a = m\frac{dv}{dt}=-v_e\frac{dm}{dt} $$

You could really say this is a parametric differential equation, and for a known thrust profile we could find $m(t)$ and $v(t)$, but we might not have that information so it's convenient to develop a form agnostic to the time variable. Algebraically, it shouldn't be hard for you to accept the following alternation, although the calculus is a little more difficult to swallow.

$$ \frac{ dm }{ dv } \frac{ 1 }{ m } = - \frac{ 1 }{ v_e } $$

If you interpret m as $m(\Delta v)$, a function of the change in velocity, then this is a differential equation with one function of one independent variable. Add in the initial condition:

$$ m(0) = m_0 $$

Then the solution is the following, which is strictly a mathematical statement about a very simple differential equation.

$$ m(\Delta v) = m_0 e^{-\frac{\Delta v}{v_e} } $$

Now that is the rocket equation.

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