Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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15c0001     Speed of a roller coaster at the top of the loop.     15c0001
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Question

 

A roller coaster has a speed of 2.0 m/s at the top of a 60.0-m tall hill as shown below. If friction is negligible, what is the speed of the roller coaster at the top of the loop that has a radius of 20.0 m?


Figure: A roller coaster.

 



Solution

 

Apply the energy principle. Define the system as the roller coaster (car) and Earth. Treat both Earth and the car as point particles, meaning that they only have rest energy and kinetic energy. And of course, there is energy due to their gravitational interaction, which is called gravitational potential energy. Also, define the initial and final states of the system. In this case, the initial state is when the car is at the top of the 60-m tall hill. The final state of the system is when the car is at the top of the loop. The energy principle states that

Substitute the energy of the system which is the sum of the particle energies and their interaction energy.

Write the particle energy as the sum of its rest energy and kinetic energy. Note that rest energy cancels out of both sides of the equation, if we assume that the mass of the car and Earth haven't changed during the process.

The only external force on the system is the force by the track. The perpendicular component of the force by the track on the car is normal (or perpendicular) to the track. The tangential component of the force by the track on the car is friction. Assume that the track is frictionless. Though this is not reasonable, it does allow us to calculate the maximum possible speed of the car at the top of the loop.

Since the force by the track on the car is perpendicular to the track, it does no work because the displacement of the car is always tangent to the track. The dot product of two vectors that are perpendicular to each other is zero. Therefore,

Substitute the work done on the system into the energy principle.

Substitute expressions for kinetic energy and gravitational potential energy. Use the approximate gravitational potential energy of a particle that is near Earth.

Define the coordinate system, showing where y = 0. Also, indicate the initial y position of the car and the final y position of the car.


Figure: The y-axis, initial y-position, and final y-position of the car.

Substitute the initial speed and initial and final y-positions of the car, and solve for the final speed of the car (at the top of the loop). Note that the mass of the car will cancel out of the equation.

The final speed of the car (at the top of the loop) is greater than the initial speed of the car (at the top of the hill). This make sense because the car-Earth system loses gravitational potential and gains kinetic energy.