I recently saw a documentary about Project Orion - nuclear pulse engine - that would reach a staggering 3-5% of the speed of light. How long would a ship take to get to Mars, Jupiter, and Saturn at that speed, assuming the spacecraft starts its journey form LEO.
3-5% of the speed of light would only be achievable for a really long duration mission. It just isn't efficient enough to do that kind of acceleration, and it takes a really long time to achieve. Okay, so how long would it take? The math is pretty easy, it's just speed/ acceleration. c=300,000,000 m/s, g= 9.8 m/s^2, so just find the speed, divide by 9.8, and you have the time it would take to accelerate that fast (Ignoring relativity, which doesn't really start to apply until 10% of the speed of light)
- 3%- 917,000 seconds.
- 5%- 1,529,000 seconds.
Okay, so both are a large time, measured in the large numbers of days. How long would it take to get to the various places? Assuming 3% c, and no acceleration time, here's what it would take:
- Mars- 25,000 seconds
- Jupiter- 88,000 seconds
- Saturn- 135,000 seconds
See that the acceleration time is much higher than the time it would take to get there. Bottom line is, a constant 1 g accelerate, and then change to 1 g decelerate would be optimal. See this question for answers to that specific problem.
It's also worth mentioning the deceleration problem. You can only go that fast one way, you can't slow down. There are tricks to slowing down using things like the magnetic field, solar sails, etc, which effectively make this possible. It's just worth mentioning to keep in mind that you eventually have to slow down.
The 3-5% isn't actually Project Orion, BTW, but a derivative called Project Daedalus. In fact, Daedalus could achieve even higher speeds. Daedalus is a bigger ship, using thermonuclear devices to achieve higher ISP, and really only works for going interstellar.