Should I let my daughters (ages 6 & 10) ride with my husband on his motorcycle?

My husband comes from a racing family. His Dad was a champion funny car driver in the 70's and his brother is a well known custom car and chopper builder. He grew up on and around dirt bikes and motorcycles and enjoys riding occasionally. I too grew up riding with my Dad on his bike as a little girl. His Dad recently ped away and he has inherited a lovely restored Triumph (not a lot of speed). When he first got the bike I didn't really think about the fact that he would want to take them for rides regularly. It was sort of a novelty, we live by the beach and the first several rides he didn't break 25 mph. Now he wants to use it more as transportation taking them to softball games, the store etc. Should I relax and let them enjoy something that both their Dad and I loved as kids or am I crazy to allow it? My friends think the latter and since times have changed so much I a regretting being so casual about it from the start. Should I leave them alone and let them bond during the occasional Sunday ride or put my foot down and forbid it?