There are a lot of things that go into mountain biking. Anyone can demonstrate physics of a bicycle just by riding one, even on straight and level ground. The gyroscopic action of the wheels can help keep you balanced while riding. But more important to a mountain biker physics can keep you from doing something too stupid.
There are different styles of mountain biking, but the one I am going to focus on is downhill mountain biking. This style of biking consists of riding downhill as the title suggests, with the help of gravity. That is the main reason why it can be dangerous. With gravity always trying to make you travel faster and faster it can be difficult to stay in control of the bike. But with the advantage of gravity many things can also be done that can't normally be achieved with out it.
For example, jumping long distances can be done much easier because of the increased momentum from descending. Better yet, the taller the mountain, the more speed and momentum that is possible to gain. This is a direct result of how high the starting point is from the base and potential energy. The higher you are the longer gravity will have to accelerate you down the mountain. So there is a potential for you to be going a maximum velocity.
Although there are many other uses of physics in understanding how riding a bike works and what can be done with it, this I felt was a more interesting topic. Bikes and kinematics aren't thought of going together a lot.