ORIGINAL: hands without shadows
If there aren't that many weeds just pull them out by hand. Racing on the track every week should keep anything from growing.
Without removing the entire plants the roots and stuff will stay and weeds will grow back eventually. You could Roundup everything and then mix a bunch of salt in with the track's dirt, but then nothing will grow in that soil for the next decade.
Roundup works grate, Used it for my buddies dirt bike track.