Edit: Actually this hurts my brain. If you're going 1,000,000MPH and fire propellant at 500MPH in your direction of travel, it would travel at approximately 1,000,500MPH and you would decelerate below 1,000,000MPH, so you actually wouldn't run into it. There's no drag on your exhaust in space like there is in an atmosphere...
For a more concrete example, for a moon rocket the speed of the rocket exhaust is around 3 or 4 km/s. On the ground the speed of the rocket is obviously 0, in low earth orbit the speed of the rocket is 7 or 8 km/s, and to initiate the transfer orbit to the moon you have to accelerate to about 10 km/s. (these would all be in earth centered, nonrotating frame speed measurements)
The rocket exhaust doesn’t have to get faster to get you to those higher speeds because you’re taking it with you.
The more you can increase the rocket exhaust relative to yourself though, the more efficient your rocket is.
Last bit of departure burn or correction burns will fly at 25km/s towards the ship, so if the ship decelerated to below that, the plume could catch up. Meanwhile, plume from deceleration continues at 75km/s away from the ship.
I was thinking that high Isp engines generally have insane exhaust velocity, like hundreds of km/s or more, that problems like this is not an issue even for interplanetary transfers. But interstellar is a bit different, depending on other factors such as dispersion, I guess?
In three dimensions though in a hard vacuum, particles coming from a fluid with a bulk velocity of kilometers per second are clearly not going to be nearby the path of the rocket for very long at all
In space, the exhaust just flies off into the distance no matter which direction you are facing.
A different but related problem: if you arrive at my space house for space dinner in your nuclear space car, it would be quite rude for you to shower me with nuclear space dust (from when you hit the nuclear space rocket brakes to stop at my house) moments before you arrive.
This isn’t an issue if you go into orbit around a planet, where the braking maneuver is at right angles to the direction to the planet surface.
Unsure. Sounds like your exhaust could enter the planet's orbit. It seems prudent to have a secondary engine for use near planets, to avoid filling their upper atmosphere with radioactive garbage.
IIRC, one of Heinlein's juvenile novels referred to this as a skew-flip maneuver; I was quite impressed reading about it at about age 12, several years after its publication date.
[Just remembered the title: Have Space Suit - Will Travel, a play on the title of the TV show Have Gun - Will Travel]
https://en.wikipedia.org/wiki/Have_Space_Suit%E2%80%94Will_T...