Question

More than anything I need 5 - 7 of this homework. 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: 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 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. 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. 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. 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. 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. 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

For the first part, the total mechanical energy of the system is conserved (no friction). The RC starts from rest at a high position. Here GPE = mgHmax = MEtot. As it descends it loses GPE and gains KE, and vice versa, but the equation GPE + KE = MEtot must be satisfied at all times. The GPE is given by the elevation of the valley or peak (GPE = mgh) so once we have this, from the previous equation of conservation of energy we can get the KE = MEtot - GPE. The velocity at a given valley or peak is given by:

Where H (75m) is the initial height and h is the elevation of the respective valley or peak.   

The momentum is equal to the mass times velocity so for each case we'll have:

Where m (400-600kg) is the mass of the RC cart.  

Next we consider the inelastic collision with another cart of equal mass that's initially at rest on a flat surface. The equation that describes the collision is the conservation of momentum:

Where v is the initial velocity of the incoming cart and V the final velocity of the two cars together, since they stick after the collision (the velocity of the two carts fused is half the initial velocity of the incoming cart). The KE's involved are:

So we can see how the total KE of the system reduces to half its initial value after the collision (KE is not conserved in inelastic collisions).

For the final part, there's a portion of track with friction that will cause the carts to stop. The equation that can be used to describe this is the work-energy theorem. The net force acting on the fused carts will be the friction force acting on the horizontal dir, since the weight and the normal force counter each other in the vertical dir (N=2mg). Therefore we can write:

Now since we want the carts to stop after the distance d (20m) then

Using the result we obtained before for KE of the fused carts after collision we get:

Where v is the velocity of the RC cart as it entered the flat surface before colliding. So the carts entered the frictional track with a certain amount of KE, that is motional energy, and this much energy is dissipated into heat and sound through friction untill the carts come to rest.