If you use continuous light, even against a well-darkened backdrop, the dart will show only as a faint streak. If the dart takes 120 ms to fly from hand to board, any given position's image will last only a fraction of a millisecond, and continuous lights bright enough to record the dart well will be huge, hot, and expensive.

A repeating electronic flash in addition to the continuous light is probably the way to go -- a series of frozen images, with the streak of light connecting them, will both give the impression of motion and tell the viewer what they're seeing.

Older camera flashes can't be set up to do this, though apparently many more current ones can. Otherwise, you need a true stroboscope, and these aren't as common as they used to be. You might be able to borrow one from a high school or college physics department, but be sure you get an understanding, up front, who pays for repairs if a 40-50 year old piece of electronics fails while you have it.

An alternative that might take several tries to get just right (and will be much harder to check success on film than digital, unless you're shooting an instant type) would be a drag shutter flash shot. If you have a shutter that can do M sync, you can fire the shutter when the dart leaves the hand, and the 20-22 ms M delay will let the dart fly partway before the flash fires, giving a single nearly-frozen (or fully frozen, depending how much flash power you need) image, but the continuous lights will still give the streak and, with a shutter speed of around 1/4 or slower, a reasonably solid image of the dart in the board.