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Voyager 1 travels at a small fraction of light speed. I've read it's fueled by hydrazine, which is a cheap combustive. Questions:

  1. What factors limit the speed of voyager 1 and similar rockets? Are there equations that sum it up?

  2. What would be required to accelerate a rocket to relativistic speeds, and how practical it is?

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  1. Fuel. Equations largely depend on the type of fuel. For example, it seems that ion propulsed spacecraft can travel much faster than conventional spacecraft.

  2. Warp Drive. We don't know if it's practical yet

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While you did answer my question, I'd appreciate some elaboration. If warp drives are necessary for relativistic speeds, why is this so? What are the theorical limits of rockets that use normal propulsion? Could you give some rough calculations on maximum speed based on types of fuel that could be used in a future (nuclear? antimatter?), and point the relevant variables? That's what I'm asking for. – Dokkat Mar 22 '13 at 3:30
Hi Raindrop. AFAIK, Ion thrusters are very much less conventional accelerators. Ion-thruster crafts take upto 2 or 3 days before they cross 500m. That's just a rough estimate. But still, they're not good at acceleration ;-) – Waffle's Crazy Peanut Mar 22 '13 at 3:40

What factors limit the speed of voyager 1 and similar rockets? Are there equations that sum it up?

Yes. There are a lot of equations, but that doesn't depend upon fuel. Rocket equation is the same (independent of fuel). The volume of fuel exhausted at the nozzle may change (difference in density of fuel, rate of emission at nozzle, etc.). This may result in a different value but AFAIK, the rocket equation remains the same. I don't know much about their fuel. But, you can check it out at Wiki for both solid and liquid propellant disadvantages.

Radioactive fuels are already in use. Nuclear rockets use a reactor to pump hot gases out. Looks like they're much better than chemical rockets.

What would be required to accelerate a rocket to relativistic speeds, and how practical it is?

No known fuel exists for that kind of question. But, we know that antimatter can be used for such Relativistic velocities. Still, it is theoretical. Maybe something would arise in the far future. But, not now..!

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The energy required to get a spaceship from point A to point B is usually referred to as the delta-v. Any Googling for info on spaceship trajectories will quickly find articles referring to the delta-v. Generally speaking we try to find trajectories that minimise the delta-v, for example by using gravity assists (as the Voyagers did).

Calculating from first principles the delta-v that a given engine with a given amount of fuel delta-v can produce is generally hard, but the values are well known for the various types of engines used in current spaceships. A quick Google should find you plenty of examples. The Wikipedia article I linked has some discussion of this; generally you increase the specific impulse by making the exhaust velocity as high as possible (without melting your engine!) or you try to increase the ratio of fuel weight to spaceship weight so you have a light spaceship and lots of fuel.

If you're specifically interested in Voyager-1 I note that the Wikipedia articles on specific orbital energy and gravity assist both mention Voyager-1. For Voyager-1 most of the delta-v came from the gravity assists and not from the engine.

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