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To see why the exhaust speed is important, let's do a calculation. Let's start with a rocket of mass $m$ going at speed $u$. (We measure all speeds with respect to some inertial reference frame.) Now, suppose it exhausts a tiny amount of propellant of mass $\delta m$ and the propellant is traveling at speed $u_P$. After it exhausts that fuel, the rocket ...


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Imagine the rocket before and after throwing a small ("infinitessimal") amount of fuel out its exhaust. You apply the momentum conservation notion by equating the increase in the rocket's forwards momentum with the momentum of the fuel thrown backwards. The easiest inertial frame to do one's analysis in is that of rocket immediately before the increment ...


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There's an accepted answer; I don't quite like it, so I'll take a whack. Suppose that the start, you don't quite know the empty mass of the vehicle, the quantity of fuel in the fuel tanks, the specific impulse, or the mass flow rate. (The empty mass of the vehicle had better be a very good estimate. Otherwise we're toast.) The space has a radar altimeter ...


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I will will assume that you fall perfectly vertical and that the celestial body does not rotate. If you assume that the mass of the rocket stays constant, then you can find when to start with the suicide burn using time reversal. Namely you "start" st the surface with your desired final velocity, $v_f$, and thrust upwards until your (specific) orbital ...


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If I am reading your question correctly, I believe you are asking whether or not you can reach outer space by simply ascending to a certain distance rather than reaching a specific velocity--that is, if by climbing in a hot air balloon you can reach a point where the gravitational potential energy is zero. If this is what you're asking, then there is a ...


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What you're describing is called an air launch.See also the Wikipedia article on air launch to orbit. As far as I know balloons are not used because they can't lift heavy enough loads. The launch is generally done using a large airliner. There is some related discussion in the question Why do space crafts take off with rockets instead of just ascending like ...


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You can think of the fuel being stored in barrels, and observers in all frames will agree on how many unused barrels are left when the ship reaches its destination, since discrete quantities aren't affected by the Lorentz boost. If you're wondering how this makes sense with the time dilation, imagine that on board the ship, it appears that the barrels are ...


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To being able to reach a high fraction of the speed of light, you need also a significant fraction of your equivalent payload mass as mass-energy. According to the formula for energy (rest and kinetic) $$ E = \gamma m c^2$$ so, to reach $0.1c$, that means your gamma is $\gamma = (1-\beta^2)^{-1/2} = 0.99^{-1/2} = 1.01$, so that means that for each kilogram ...


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The spaceship and the Earth start out in the same frame, so the amount of fuel agrees in both frames.


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The rocket equation is a cruel tyrant. By far the hardest challenge is just finding a propulsion system that can achieve the necessary velocities. Project Daedalus proposed to use an inertial confinement nuclear fusion system, but we really don't know if that can be made to work. The best existing inertial confinement technology is the National Ignition ...



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