The High-Speed Physics of Olympic BMX

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http://interrete.org/wp-content/uploads/2016/08/spring_2016_sketches_key10.jpg

There’s a lot happening at the start of an Olympic BMX race. Athletes begin at the top of a ramp, which they descend while pedaling and being pulled by gravity. At the end of the ramp, they transition from pointing down to aiming horizontally. You may not think there are many physics problems here, but there are.

How fast would you go if you didn’t pedal?

One claim about Olympic BMX is that the riders descend the ramp in two seconds at a speed of about 35 mph (15.6 m/s). What if you simply rolled down the incline and let gravity accelerate you? How fast would you be going? Of course, this question depends upon the dimensions of the ramp. An official starting ramp has a height of 8 meters with dimensions something like this (they aren’t completely straight).

Spring 2016 Sketches key

Instead of a bike, I’ve placed a frictionless block at the top of the ramp. If I want to determine the speed of this sliding block at the bottom of the ramp, I can start with one of several principles. However, the Work-Energy Principle is the most straightforward approach. This states that the work done on a system is equal to the change in energy.

La te xi t 1

If I view the block and the Earth as the system, the only external force is the force from the ramp. This force always pushes perpendicular to the direction the block moves such that the total work on the system is zero. That leaves a total change in energy of zero Joules. In this case, there are two types of energy–kinetic energy and gravitational potential energy.

La te xi t 1

There are two important points about gravitational potential energy:

  • The value of y doesn’t really matter. Since the Work-Energy Principle only deals with the change in gravitational potential energy, I only care about the change in y. For this situation, I will use the bottom of the ramp as my y = 0 meters (but you could put this anywhere).
  • Again, the change in potential only depends on the change in height. It does not depend on how far the block moves horizontally. This means that the angle of the ramp doesn’t really change the final speed of the block (but only in the case where friction doesn’t matter).

With this in mind, I will call the top of the ramp position 1 and the bottom position 2. The Work-Energy equation becomes:

La te xi t 1

Since the bikes start from rest, the initial kinetic energy is zero. Also, the final potential energy is zero since I set my y value at zero at the bottom. Here I am using h as the height of the ramp and the initial y-value. Now, I can solve for the final velocity (the mass cancels) and get:

La te xi t 1

Using a height of 8 meters and a gravitational constant of 9.8 N/kg, I get a final speed of 12.5 m/s–slower than the 35 mph as stated above. Actually, a real bike would have an even lower speed for two reasons. First, a frictional force would do negative work on the system. Second, bikes have wheels that spin. When a wheel spins, it requires extra energy to make these wheels rotate such that some of the change in gravitational potential energy would be used for rotation instead of translation.