Physicist: In the post “Q: Have aliens ever visited Earth?” I said that the maximum velocity that can be obtained by a rocket using fusion is 11% of light speed. Wrong!
It turns out that (as a commenter had suggested) the maximum velocity of a rocket is actually a function of the exhaust velocity of the rocket (which is about 11% of light speed for fusion), and the ratio of initial to final mass of the rocket (the initial mass includes fuel). The assumption I had been working with was that a rocket shouldn’t be able to get going any faster than it’s exhaust velocity.
To find the mass ratio in question: where Mi is the initial mass, Mf is the final mass, c is the speed of light (2.99 x 108 m/s), w is the exhaust velocity, and V is the final speed of the rocket. A derivation of this can be found here: www.relativitycalculator.com/images/rocket_equations/AIAA.pdf.
You can also turn this around: , where .
A “reasonable” rocket would be something like 99% hydrogen fuel. Which translates to a final velocity of only about 47% of light speed. In order to get something to 0.9c (again using fusion) would require a fuel-to-ship ratio of around 700,000. If you wanted to get the shuttle to 0.9c you’d need about 19 cubic km of liquid hydrogen. Frustrating, isn’t it?
Using normal rocket fuel (w = 4440 m/s) to get to 0.9c, you’d need a fuel to ship ratio of about 1076,985. So if the space shuttle were to use the engines it has now, it’s fuel tanks would weigh at least as much as about 1076,961 Suns.