I was doing some back-of-a-postage-stamp calculations on high-isp interstellar spacecraft earlier and realised something. If I design a spaceship which according to the rocket equation has a very high Delta-V, close or even beyond the speed of light (in my calculations it was 6.7 million km/s or 22c), how do you calculate what speed that spacecraft can actually accelerate to? Obviously a spacecraft with a Delta-v of 6.7 million km/s cannot actually accelerate to that speed (with it being faster-than-light and all), but how fast could it get?
For those wondering, here’s my math on the example craft I referenced:
Assuming photon drive powered by 100% efficient antimatter reactions, 1 exawatt of power to the drive = 3 million kilonewtons of thrust or 337,000 tons thrust. This requires 1.7 million tons of antimatter (& 1.7 million tons of matter, 3.4 million total), and can fire for 10 years.
Assuming mass fraction of 90%
TWR of ~0.08
Delta V of ~6.7 million km/s (22c)