We have a Motocross course near our village, and they hold national events there (very noisy) about six times a year. The Motocross bikes look like mopeds, but they probably aren't, as they zoom around the track like anything. It's extremely hilly (artificial hills of course - Norfolk is flat!). Folk here call them the Scramblers.

I think the clue to a moped is the 'ped' bit. Originally they had pedals with which to start the motor and to assist it to get up inclines etc.