Very distant objects need very little speed to orbit, take for instance a geosynchronous orbit that orbits every 24 hours, which enables it to hover over the sa

Very distant objects need very little speed to orbit, take for instance a geosynchronous orbit that orbits every 24 hours, which enables it to hover over the same part of the equator indefinitely. With a braking burn from geosync orbit your orbital period would keep decreasing from 24 hours as your orbital velocity keeps increasing. The ISS and similar low earth orbits have a period of 90 minutes. Keep in mind if trying to speed up/catch another object you can just use more thrust and keep adjusting your angle. But it's fuel intensive and when you stop thrusting you might well be in a crazy orbit, very different than the target you are trying to catch. There's a game called Osmos on most platforms that turns this kind of thing into a game, I recommend it. As mentioned elsewhere there's kerbel space program, however there's much more than orbital mechanics involved.