Question

You have been asked by your supervisors at A&L Engineering to design a roller coaster for a new theme park. Because this design is in the initial stages, you have been asked to create a track for the ride. Your coaster should have at least two peaks and two valleys, and launch from an initial height of 75 meters. Each peak and valley should represent a vertical change of at least 20 meters. In your design, you should plan for a mass between 400 and 600 kilograms.

Once you have designed the track, you have been asked to calculate the kinetic energy, potential energy, momentum, and work done by the cart at various points throughout the track. Unless otherwise stated, you can ignore the effects of friction. Following your calculations, you have been asked to describe the energy transfers detailed by these equations.

Directions

To complete this roller coaster design report, complete the following:

1. Create a diagram of a roller coaster track containing at least two peaks and two valleys. As you complete your report, you may wish to design a more complicated coaster. However, it should still have two peaks and two valleys that meet the requirements below and that you are comfortable using in calculations and descriptions of energy and momentum. Your diagram should include the following information:
   • An initial height of 75 meters
   • At least two peaks and two valleys representing drops of over 20 meters
   • A set mass for your roller coaster cart between 400 and 600 kilograms

2. Calculate the kinetic energy, potential energy, and momentum of the cart at the initial drop for both peaks, and for both valleys. If your coaster has more than two peaks and two valleys, select which peaks and valleys you wish to use in your calculations and clearly mark them on your diagram. In your calculations, be sure to explicitly state the equations you use and what values you will be substituting to calculate the final value.

3. Describe the energy transfers that occur as the cart moves along the track. This should be a narrative description of the energy transfers that occur at the initial launching point, peaks, and valleys. In your descriptions, address the following:
   • At each of the identified points, how was kinetic energy transferred to potential energy, and vice versa?
   • What happens to the total energy of the cart as it moves along the track? Why?
   • How is the principle of conservation of energy applied in this situation?

In addition to your description of the motion of the cart on the track, you have been asked to model the motion of the cart as it comes to a stop at the end of the coaster. For these calculations, assume that the cart will inelastically collide with a cart of equal mass at rest on a flat surface.

4. Calculate the momentum and kinetic energy of the cart before and after an inelastic collision. In your calculations, be sure to explicitly state the equations you use and what values you will be substituting to calculate the final value.

5. Describe the energy transfers that occur as a cart inelastically collides with an object of equal mass at rest. This should be a narrative description of the energy transfers that occur as the cart inelastically collides with a cart of equal mass. In your descriptions, address the following:
   • What was the kinetic energy of each cart before and after the collision?
   • What happens to the total energy of the system, now including both carts, as a result of the inelastic collision?
   • Describe how the principle of conservation of energy is applied in this situation.

Following the inelastic collision of the carts, the two carts fuse into an object with double the mass of the original cart. There is then a frictional section of the track to slow the cart to a stop over 20 meters. Describe the amount of work due to friction and frictional force exerted to stop both carts over 20 meters.

6. Calculate the work due to friction and frictional force. In your calculations, be sure to explicitly state the equations you use and what values you will be substituting to calculate the final value.

7. Describe the energy transfers that occur as the cart is brought to a stop. This should be a narrative description of the energy transfers—written to describe these concepts to a nontechnical audience—that occur as the cart is brought to a stop. In your descriptions, address the following:
   • What is the kinetic energy of the cart system before and after it has been brought to a stop?
   • What happens to the total energy of the system as a result of this change in motion?
   • Describe how the principle of conservation of energy is applied in this situation.

Answer 1

The Design:

The design has 2 peaks P1 and P2 and 2 valleys V1 and V2 (more like troughs than valleys, you can elongate these parts if you want). The launching point is A.

At A,P1 and P2, the potential energy is very high and the kinetic energy is very low. At the valleys V1 and V2 the kinetic energies are very high and the potential energies are low.

Let the mass of the cart be 500 kg (M).

Energies and Momenta at the peaks and valleys:

The potential energy at A:

This energy is purely potential since the cart starts from rest.

When the cart drops to V1, the potential energy lost=kinetic energy at V1

Let's say that the velocity of the cart at V1 is

Therefore, the kinetic energy is:

Therefore, the velocity:

Momentum at V1

The potential energy at P1:

By conservation of energy, the total energy at A is the same as the total energy at P1.

In-fact, we are assuming that the total mechanical energy (KE+PE) is same everywhere on the track except for the end where we need to introduce friction.

Therefore, the kinetic energy at P1:

Velocity at P1:

We will do the same for V2 and P2

Collision and Stopping:

In the flat region at the end,

Therefore, the kinetic energy:

which is the same as

Therefore,

The momentum:

Momentum is conserved in collision. Also the collision is inelastic.

since the carts join to form a single entity

Therefore,

This velocity needs to be reduced to 0 by friction

Therefore, force of friction:

Conservation Laws and Energy Transfer:

Conservation of Energy: The total mechanical energy of a system remains constant in the absence of dissipative forces such as friction or air resistance. This happes throughout the track.

Energy transfer: Over the track, the energy is transferred from Kinetic to potential and vice versa as it travels through troughs and valleys. At the end, when friction comes into play, the energy will be converted to heat between the track and the wheels of the cart and sound(if any).

Conservation of momentum: In a collision, there are no external forces, therefore, the total momentum before and after the collision remain constant.