Alternative Homework Assignment: Roller Coaster
Designing a roller coaster requires careful consideration of several physics concepts. Most notably, mechanical energy must be kept track of to ensure the Roller Coaster completes the course. Friction is also an important consideration, however it is rather complicated to deal with exactly. A simple but poor approximation can be made by assuming a constant drag force resulting in a constant energy dissipation per unit track length. This is not very accurate as shall be seen later, but it is better than ignoring friction all together.
Another important design consideration is the accelerations experienced by the passengers. The goal here is to maximze the excitement of the ride without actually endangering the passengers. An important figure to remember here is that an average person will black out if the "headward" acceleration s/he experiences reaches 2 g. By comparison, people can withstand accelerations on the order of 10 g in directions perpendicular to their bodies without any permanent adverse effects, however most Roller Coaster cars would jump their tracks at such extreme accelerations.
a) The object of this problem is to design a roller coaster with at least the following features. 1) A large hill with a chain lift on one side that is the source of energy. 2) A vertical loop or "loop the loop". 3) A horizontal loop. 4) A series of at least three small hills. You may include more features in your roller coaster if you wish. Begin your roller coaster design by choosing the ordering of the features and use your experience with roller coasters to make some rough estimates of the dimensions of the features.
b) Now make the plan a little more exact. On graph paper, make a scaled drawing of the track from the side in such a way that if you connected the left side of the paper to the right to make a cylinder, you would have a reasonable three dimensional picture of the track. In other words, even though the track makes a large loop in reality, draw it as if the train started on one end of the paper and ended on the other. Note that you will need to take special care at the horizontal loop(s) to be sure the length of track proposed is sufficient for the size loop(s) you intend. Use this drawing to calculate the total length of the track for your roller coaster.
c) What about the train for your passengers? Assuming it consists of five cars with a capacity of four people per car, estimate the total mass of the train and its length.
d) Next use your knowledge of physics to see which parts of your rough design work, and what needs to be altered in the parts that don't. The first check is to make sure that the train has enough energy to reach the end of the track. The Roller Coaster is powered entirely by the gravitational potential energy it receives in being pulled to the top of the first hill. For this reason, a good first check of the Roller Coaster is to make sure that the first hill is the tallest feature on the ride. If this is not the case on your drawing, choose dimensions that fix it, but do not draw a new picture yet.
a) We also need to estimate the energy dissipated by frictinal drag on the train for each meter of track it traverses. As a lower limit on this frictional energy loss let's consider one of the tallest and fastest roller coasters in the world. It has a height of 62.5 m and a top speed of 34 m/s. This speed is reached at the bottom of the first hill which has a slope of 60. Comparing its actual speed at the bottom of this hill to that achieved in the frictionless case, estimate the average energy dissipated per meter of track going down this hill. Take this to ba a reasonable minimum rate of energy dissipation.
b) At the end of the ride, the Roller Coaster will need to be stopped without accelerating the passengers too violently. For this to be the case, the ride will have to dissipate enough energy for the final speed to be about five meters per second. Use this final speed, the height of your track's first hill, and the total length of your track to estimate a maximum rate of dissipation. Use the two rates you have calculated to make a reasonable estimate of the rate of dissipation for your roller coaster.
c) With an estimate of the rate of dissipation now in hand, it's time to check that your roller train can rally make it up to and over each of the highest points on your roller coaster. Measure the amount of track the train has to travel over to reach each of those points. From these data, calculate the amount of energy the train has at each of these points. If you find that there is any place where the train is higher than the available energy allows, then rescale, and/or reorder the features so that the train is able to reach every point on the track.
a) Now consider the vertical loop in some detail. In order to stay on the track without depending on the wheels beneath the track, the train must experience a downward, centripetal acceleration at the top of the loop that exceeds 1 g. The train's speed is determined by its height and the amount of track it has traveled across. Considering this, what should the maximum radius of your track's vertical loop be.
b) In the frictional case, the point were the train is most likely to fall is not the top of the loop, but the point where the component of the train's weight in the direction of travel is equal in magnitude to the frictional force. Why is this the case, and where is this point on your track? What should the new radius of the loop be?
b') (optional) The above is a case where the constant friction approximation is particularly poor. If the air friction is considered to be negligible, then the frictional force should be proportional to the normal force. The normal force in the loop is a function of the train's position and speed. If the average friction used above is the frictional force on level ground, then where is the new critical point?
c) The above analysis is fine for the train's center of mass, but is not accurate for the first or last car. The reason for this is that the entire train must always have the same speed which is determined by the center of mass. The passengers in the middle car experience essentially this motion. The passengers in the front and back however have a slightly different experience. Calculate the train's speeds when the first and last cars are at the top of the loop. From this calculate the acceleration felt by the first and last car's passengers. What should the loop radius be if all of the cars are to stay on the track? (If you really wanted to treat the friction accurately in this problem you would have to consider that each car has its own normal force and that all of them contribute to the overall friction.)
d) Calculate the acceleration experienced by the passengers of the first, middle and last cars at the bottom of the loop. Is this reasonable?
e) If the bottom of the loop is causing your passengers to black out, the acceleration may be reduced by cutting the circle of the loop into four quarters and replacing the bottom two quarters with sections of a much larger circle. Design such a loop that takes the track from horizontal back to horizontal without causing any of the passengers to black out. Make sure the train will still stay on the track.
a) Consider the other features of your Roller Coaster one by one for safety. The best way to do this is to find the place with the greatest acceleration and make sure that it is safe for the first, last, and middle cars.
b) As a last touch, find the slope of the bank for the horizontal loop. You can do this by making sure the force on the passengers is always perpendicular to the track. If your loop dimensions allow it, have the track sloped so as to keep the velocity constant throughout the horizontal loop. This should make your calculations easier.
c) Use all of the adjustments you have made to make a final draft of your Roller Coaster drawing. Make sure that the train can reach all of the critical points with appropriate velocities. Calculate the maximum and final velocities speeds of the train as it completes a cycle.
d) Explain what car you would prefer to ride in and why. (The answer to this might depend greatly upon your personal feelings about roller coasters.)
"g" as a unit of acceleration refers to the gravitational acceleration near the surface of the earth (9.8 m/s2). This unit is often used when referring to accelerations experienced by a person in a vehicle because when the vehicle accelerates, it pushes the passenger along with it. This pushing changes the normal force experienced by the passenger and the person feels as if s/he is in a gravitational field that is the resultant of the actual gravitation field and the negative of the vehicle's acceleration. For example if the vehicle were accelerating in the up direction at 1g, the passenger would experience 2g of acceleration and pass out while if the vehicle were accelerating down, the passenger would feel weightless.
These problems written and collected by K. Vick, E. Redish, and P. Cooney. These problems may be freely used in classrooms. They may be copied and cited in published work if the Activity-Based Physics (ABP) Alternative Homework Assignments (AHAs) Problem site is mentioned and the URL given.
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