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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 ...

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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 ...

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Such a dramatic change would indeed make it harder to reach space, in a precisely defined sense. The easiest measure of how 'hard' it is to get so space is the escape velocity, which is determined by the planet's mass $M$ and radius $R$ as $$v_\text{esc}=\sqrt{\frac{2GM}R}.$$ For a planet like the one you mention, the escape velocity goes up by a factor of ...

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It never becomes impossible per se, but at some point there could be so much gravity that construction of a working rocket would be beyond our current ability to engineer something that could work. That is, it might take impracticably huge quantities of fuel, or require materials stronger than we can construct. There are just a couple of amazingly simple ...

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"I've been pondering the implications of time dilation. " - thar be dragons! :) The answer to the question is no, of course. Per the previous answers you have to consider frame of reference which in space time is arbitrary. Time dilation is how you resolve the problem of two objects traveling directly towards each other at > .5C per a 'stationary' third ...

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that to an external observer it would appear to be moving very much slower? I don't understand the reasoning here. When you write assume that if a craft was travelling at a speed very close to the speed of light I take that to mean that the craft is travelling very close to the speed of light according to an external observer. Keep in mind ...

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No. Time-dilation is the slowing of time as experienced by the fast moving craft, not the 'stationary' observer. Remember that light moves at c, and we see it move at c, not some slower or stationary speed. As the craft approaches c, it appears to accelerate increasingly slowly; from 0.99999c to 0.999999c is only a difference of 2.7 km/s, but it is still ...

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In the idealised case, the answer to this is slightly surprising. The fact that the mass of a rocket must include the mass of its fuel is embodied in the rocket equation, $$\Delta v = v_e \ln\frac{m_i}{m_f},$$ where $m_i$ is the initial mass of the rocket (including fuel, payload and everything else), and $m_f$ is the final mass, including the payload but ...

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The upper limit is the size of the planet. Consider that you could mount a linear accelerator or "mass driver" pointing upward and use the material the planet is made of as reaction mass. Now drive off to explore the cosmos. By the time you whittle down the planet to an asteroid sized rock, you will be going at a pretty good clip!

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This is an optimal control problem, so I will use the rules of optimal control. First, we represent the state space equations. Also we take the total mass as a state and amount of fuel burnt as the input control. So we have: \begin{cases} \tag{1} \dot{x}_1=x_2 \\ \dot{x}_2=\frac{\eta \theta-k(x_1)x_2^2}{x_3}-g \\ \dot{x}_3=-\theta \end{cases} with these ...

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I thought this question was interesting and I didn't want to do any proper work this afternoon so I made a simple model to find out what would happen. My matlab code is at the end of the question. So far I've tested three cases and considered changing the initial thrust and adding a linear increase in the thrust for each case. The thrust is given as a ...

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The quantity that tells you what time an observer travelling along a path $\gamma : [t_0,t_1] \rightarrow \mathbb{R}^4$ experiences is the proper time $$\tau = \int_\gamma \sqrt{\mathrm{d}x_\mu\mathrm{d}x^\mu}$$ Assuming flat spacetime, i.e Minkowski metric/special relativity, this reduces to  \tau = \int_\gamma\sqrt{\mathrm{d}t^2 - ...

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Rockets work better in a vacuum, for two reasons: Thrust is higher and drag is lower (non-existent) in a vacuum. The drag issue is obvious, and also a bit off-topic. The question is about thrust. A bit overly simplistic explanation is that there's nothing to impede the exhaust in a vacuum. For a given fuel, rocket exhaust velocities are typically 15% to 20% ...

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