The Bicycle Brake and the Steep Hill
A student rides a bicycle down a hill 120 m long that drops 15 m in height. The bicycle and rider have a combined mass of 80 kg. At the bottom the student applies the brakes and stops in 8 m. The brakes consist of rubber blocks pressing on the steel wheel rims. The student notices the rims become hot and that braking is poor after riding through a puddle. She wants to know how much energy the brakes must absorb and why wet brakes are less effective. Take g = 10 N/kg.
1. Calculate the gravitational potential energy lost on the hill and the average braking force needed to stop in 8 m, stating any assumptions.
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Potential energy lost: Ep = mgh = 80 kg x 10 N/kg x 15 m = 12 000 J.
Assumption: Air resistance and rolling friction are ignored, so all of this energy becomes kinetic energy at the bottom of the hill. In reality some is lost to air resistance and tyre friction, so the kinetic energy at the bottom is somewhat less than 12 000 J. The calculation gives an upper limit.
Kinetic energy at the bottom: about 12 000 J (upper limit).
Braking force: The work done by the brakes equals the kinetic energy removed. Work = force x distance, so F = E/d = 12 000 J / 8 m = 1500 N.
Comment on the answer: 1500 N is about twice the rider's weight (800 N), which is large but plausible for an emergency stop from speed. A longer stopping distance reduces the force: stopping in 16 m would need 750 N. This is why braking gently is safer, especially on a wet rim.
Where the energy goes: It is transferred to thermal energy in the brake blocks and rims. The temperature rise depends on the mass and specific heat capacity of the rims and blocks; steel rims have a fairly low specific heat capacity, so they heat quickly, which matches the student's observation.
2. Explain why the brakes work poorly after the puddle, and why the rims get hot. Suggest one design or riding change that would improve safety.
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Why wet brakes are poor:
- Water film: Water between the rubber block and the steel rim acts as a lubricant, reducing the friction force. With less friction, the braking force for a given squeeze is smaller, so the stopping distance increases.
- Cooling: The water also carries heat away and can turn to steam, which can form a layer between block and rim and further reduce contact.
- Heating: As the brakes do work against friction, kinetic energy is transferred to thermal energy in the rims and blocks. The rims get hot because they have a relatively small mass and a low specific heat capacity, so a given amount of energy raises their temperature a lot. This is consistent with the 12 000 J calculated: that energy has to go somewhere, and most of it goes into the rims and blocks.
Design or riding change:
- Riding: Brake earlier and more gently on wet descents, using both brakes, and test the brakes after a puddle before needing them. Reduce speed before the hill so less energy must be dissipated.
- Design: Use brake blocks made of a compound designed for wet conditions, or disc brakes where the braking surface is shielded from splashes and water is thrown off by rotation. Regular checking of block thickness and rim wear is also important, since worn blocks and thin rims brake poorly and can fail.
Judgement: The puddle does not change the energy that must be removed; it changes the friction available to remove it. The safe response is to allow a longer stopping distance, not to squeeze harder on a wet rim.