Coaster Dynamics

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Work and Friction

To prevent the car from going too fast at the end of the track, we need to raise the height of the end point, so that all of the car's initial potential energy is consumed by the work done by friction. The relationship between friction and the speed of the roller coaster car derives from the Equivalence of Work and Energy. A fundamental principle of physics is the Conseravtion of Energy. This principle states that the total energy of an object (like a roller coaster car) is constant, although it can switch from one form to another. For a roller coaster, the two most important types of energy are potential energy (the energy resulting from gravity) and kinetic energy (the energy of motion). This principle holds true for a "conservative system" -- one in which the only forces acting are "conservative" ones (like gravity). However, some forces do not conserve energy -- and one of the most important "non-conservative" forces is friction. Unlike gravity, the force of the friction of the wheels in contact with the roller coaster track causes energy to be lost. The energy that is lost is converted mainly into heat and noise. The principle known as the "Equivalence of Work and Energy" says that the lost energy is equal to the work done by the friction. The term "work" has many meanings, but in physics, work can be done by any kind of force, and is defined as: Work = (F)(L) F = Force (N) L = Displacement (m). Consider the case of the roller coaster car going up and down the track, subject to a friction force, as illustrated in the diagram.

For the up and down motion, energy is conserved -- as the energy is exchanged back and forth between potential and kinetic energy. The potential energy is equal to: PE = mgh m = mass (Kg) g = gravity constant (9.82 m/s²) h = height (m).

The kinetic energy is equal to: KE = (1/2)(mv²) m = mass (Kg) v = velocity (m/s).

In addition, the energy consumed by the non-conservative friction force is equal to: W = Friction Work = (F)(L) F = Friction Force (N) L = Track Length (m). Therefore, at any point on the track, the energy of the roller coaster car is equal to: Starting Energy - W = PE + KE. For the "rolling friction" of a wheel, the friction force is roughly constant, regardless of the speed of the car. Thus, for the roller coaster car, the total Friction Work is roughly constant (as long as the track length does not change very much). In the Coaster Dynamics simulation, friction is not the only non-conservative force included. In the computer calculations, the force caused by wind resistance is also included. The work done by wind resistance increases with increasing speed. However, for simplified calculations, the wind resistance can be neglected.

Apply Your Knowledge

Your Task:

Knowing the final velocity of the roller coaster car, what should the height of the track end point be so that the final speed of the car will be nearly zero?

What We Know:

The known parameters are: mass of the car = 453.59 kg g = 9.82 m/s² h = height of the first hill = 62.50 m.

Using the "Coaster Lab" Option in Coaster Dynamics, determine the values of the following parameters: v = final velocity (m/s) H1 = height of the track end point (m).

For our simplified analysis, we will assume that the speed of the car at the start of the ride at the top of the first hill is zero. In that case, the only starting energy is potential energy, which is equal to: 1)   Starting Energy = mgh. At the end of the track, the total energy is equal to: 2)   Ending Energy = (mg)(H1)+(1/2)(mv²) H1 = height of the track end point. The difference between the Starting and Ending Energy is the Friction Work. Thus: 3)   mgh - W = (mg)(H1) + (1/2)(mv²).

What To Do:

If we changed the height of the track end point to H2, to satisfy our goal of zero velocity at the end of the track, the final kinetic energy would be zero. Thus, the Starting Energy would have to equal the energy consumed by the Friction Work: 4)   Starting Energy = (mg)(h-H2) = W H2 = New height of the track end point.

Assuming that the Friction Work remains constant, using equations 3) and 4), and the known parameter values from above, calculate H2, the new height of the track end point. To test your work, change the height of the track end point to your calculated value, and then re-run the roller coaster ride. Remember that you did not include the affect of wind resistance, and the Friction Work is actually not quite constant since the track length will change slightly. So, the answer you calculate will not be exactly correct.