The Exact Maximum Speeds of Different Roller Coasters Continuous Discree
Calculus in Roller Coasters!
By Michael Cappella and Kirsten Haltman
Amusement park rides use physics and applied calculus to provide a thrilling experience for the rider. Roller coasters do not have their own engine; rather they rely on the conversion of potential energy to kinetic energy. At the beginning of the ride, the roller coaster car is pulled to the top of the first hill, but after that the coaster must complete the ride on its own. During the course of the ride, different kinds of wheels help to keep the ride smooth; running wheels guide the coaster on the track, friction wheels control lateral motion, and a final set of wheels keeps the coaster on the track, even if the cars are inverted. Special compressed air brakes stop the car as the ride comes to a stop.
Roller coasters originated in Russia in the 1600s, where the earliest forms were wood-framed ice sleds, and they developed the essential principle of friction using sand, paving the way for other countries to follow. The first under friction roller coaster came in 1912 on Coney Island in New York, and the first national theme park came in 1955. This theme park, Disneyland, contained the first tubular roller coaster, the Matterhorn Bobsleds. As mathematics advanced, roller coaster development became continuous, allowing roller coasters to evolve from the small ones at Disneyland to the massive ones we know today.
Calculus and physics are directly related to roller coaster design. There are two types of energy used in roller coasters: potential energy, and kinetic energy. The potential energy is the amount of energy available for the coaster to have. It is calculated by the equation PE=mgh where m=mass, g=gravity, and h=height. Therefore at the top of a hill, a roller coaster's potential energy is at a maximum, and at the bottom of the hill or the lowest part of the ride, it is at a minimum. The second type of energy is kinetic energy or the energy of motion. It is calculated by the equation KE=.5mv^2, where m=mass and v=velocity. Kinetic energy is at its maximum at the bottom of a hill and the minimum is at the top of the hill. Energy can not be created nor destroyed so over the course of the ride, there is a trade off between the amount of kinetic energy and potential energy. Total energy remains the same throughout the ride.
We can represent a mock roller coaster by using the function f(t)= sin(t), where t=time from 0 to the time it takes to complete the ride. By simply plugging in any time t into the function, we find the position of the roller coaster car at that time. If we were to take the first derivative, f'(t)=
[sin(t)]'= v(t)= cos(t), and plug in any time t, we would solve for the velocity of the roller coaster at that time. The second derivative, f"(t)=v'(t)= [cos(t)]'= -sin(t), finds the value of the acceleration of the roller coaster at time t. Acceleration can be negative or positive. The third derivative, a'(t)=[-sin(t)]'= -cos(t) is jerk, which affects comfort.
This relatively simple concept of calculus plays a huge role in the design and construction of roller coasters today. Obviously every roller coaster is defined by a different function, much more complex than sin(x), however through this example we learn about how the velocity and acceleration at any given point affects the ride.
The biggest thrill of roller coasters is the loop, also known as a clothoid loop. A clothoid loop has a constantly curving shape, with sections that resemble the curve of a circle. As they go through the loop, a coaster rider is constantly changing his/her direction of motion. At all times, the direction of motion could be described as being tangent to, and this change is caused by the presence of unbalanced forces, which results in an acceleration. However, this acceleration is constantly changing, and normally changes as the rider moves from the top of the loop to the bottom of the loop. These constant and drastic changes are also the cause of much of the thrill, as they feel similarly to how gravity acts on your body.
The final force acting on your body is inertia. When the loop goes upside down, it's inertia that keeps you in your seat. Inertia is defined as the force that presses your body to the outside of the loop as the car spins around the loop. Although gravity is pulling your body towards the earth, the acceleration at the very top of the loop is stronger than gravity and pulling your body upward, counteracting gravity. However, the loop must be elliptical, otherwise centripetal (g) force would be too strong for safety and comfort.
Batman the ride at Six Flags in California includes several common roller coaster designs; it has sharp turns, zero gravity, inversions, and incredible speed. We can calculate the total energy at a specific point through calculating both the kinetic and potential energies, and then adding them together. At the lower point where the tracks cross in the picture to the right, the total energy is 17,726.25 total Joules of energy.
Source: https://thecalculusofrollercoasters.blogspot.com/2014/12/the-calculus-of-roller-coasters.html
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