DrPizza -
No, you've got it wrong. The capstan and rubber pinch roller next to the head always spins at the same speed; it sets the speed of the tape past the head, and that is a constant. The Take-up reel is driven with a slipping clutch so that it maintains a specific tension on the tape, but spins at precisely the speed necessary to pick up the tape that the capstan is driving. The next time you play with a cassette player, stick something into the hub of the takeup reel and stop it; you'll find that the tape continues to move past the head, and the feed reel will continue to move, until the tape gets sucked into the mechanism and stops everything.
dkozloski -
No, four wheel drive vehicles DO have the same final drive ratios front and back. The simplest transfer cases hard-connect the front and back driveshafts which can lead to some humongous torsional stresses on the driveshafts and components as the car turns on hard surfaces (the rear wheels travel a shorter distance than the front). All-wheel drive vehicles solve the problem in various ways - they can turn the transfer case into a differential (in which case the nomenclature "4 wheel drive" is doubly incorrect), they can put a clutch into the transfer case allowing a limited-slip action between the front and rear, or any of several other methods. Having different final drive ratios front and back would imply that, in four-wheel-drive, one set of wheels would always be spinning, in other words, using dynamic friction rather than static friction. Not such a good idea on ice or other slick surfaces.
And Logic1485, I think you've just not done enough observations. If you play a tape starting at the beginning, the take-up reel has no tape on it, and thus a smaller diameter, than the feed reel. It will spin faster as a result. Halfway through the tape, they should be spinning about the same speed. Near the end of the tape, the take-up reel will be spinning slowly, and the feed reel will be spinning fastest.
I have a VCR that uses this principle to know when it is getting near the front/end of the tape in it's hyperdrive rewind/fastforward modes. It measures the difference in RPM between the take-up reel and the feed reel, and can calculate the difference in diameters between the two. About 2 minutes before either end, it slows down and waits for the tape tension to change (or the clear leader to appear) to tell it when to stop.
/frank