Calculating the Height at Which a Bike Comes to Rest

What is the process to calculate the height at which a bike will come to rest?

How can we determine the height when a bike rider approaches a hill at a speed of 8.5 m/s with a combined mass of 85.0 kg?

Height Calculation Process

When a bike rider with a mass of 85.0 kg and a speed of 8.5 m/s approaches a hill, we can calculate the height at which the bike will come to rest by considering the conversion of kinetic energy to potential energy.

The rider and the bike have a combined mass of 85.0 kg and a speed of 8.5 m/s. As they coast up the hill with no friction, all of their energy is initially kinetic energy. This kinetic energy will be converted into potential energy as they climb the hill.

By using the conservation of energy principle, where the sum of potential and kinetic energies remains constant, we can set up an equation:

Potential Energy + Initial Kinetic Energy = Final Kinetic Energy + Final Potential Energy

Since the final kinetic energy and the initial potential energy are zero at the moment of rest, we have:

Potential Energy = Initial Kinetic Energy

The formula for potential energy is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. The formula for kinetic energy is 1/2 mv², where m is the mass and v is the velocity.

Setting the potential energy equal to the kinetic energy, we get: mgh = 1/2 mv²

Solving for h, the height at which the bike comes to rest, we find: h = v²/2g

Substituting the given values, we calculate the height to be 3.69 meters.