@Steve, coding a game because you want to make something good and enjoy making it is the best reason, and likely will result in something well designed and worth playing for everyone.
What you say about the cops reminds me of an awsome old amiga game called road rash. You played a biker racing against other bikers, however this wasn't exactly the formula one.
Firstly you could punch and kick your opponents off their bikes. Kicking knocked them streight off, but slowed your speed down while it took three punches to knock someone off. If you were knocked off your bike, you had to go and pick it up again (and obviusly you walked far slower than your bike).
If the cops turned up, they'd hall you in if you were speeding and end the race so you had to be careful, you also got some cops who would try to smack you off your bike with their night sticks and if they got you off they'd haul you in.
You could also get hit by oncoming traffic and knocked off your bike as well.
One thing I particularly liked is that all the other bikers in the race and the cops had personalities, and you'd always see in your rear view mirror the name of who was approaching or who you were coming up to, and after the race everyone had quotes, my favourite was the one who said "I'm going to be cleaning my bike with your face!"
Apart from all this it was a normal racing game, accept that having a motorbike you obviously had a lot more room on the road to play with. You could buy better parts for your bike with the money you got racing,a nd eventually better bikes as well.
All in all the game was a lot of fun, I enjoye the beatemup aspects rather more than the racing, indeed i never used to bother about winning races I'd just see how many other bikers I could brutalize .
Btw, I'm not suggesting you build a game like this, just sharing experiences sinse Burak's comment about the cops reminded me of Road rash, plus this was an example of a racing game I did! have fun with despite my usual preferences.
With our dreaming and singing, Ceaseless and sorrowless we! The glory about us clinging Of the glorious futures we see,
Our souls with high music ringing; O men! It must ever be
That we dwell in our dreaming and singing, A little apart from ye. (Arthur O'Shaughnessy 1873.)