The rides in an amusement park not only are fun but also demonstrate principles in physics. Some of these principles are inertia, acceleration, and energy states.
Inertia plays a significant role in the physics of a roller coaster. Inertia tends to keep the roller coaster moving foward along the track at a certain speed, even when nothing is pushing on the roller coaster. While a roller coaster will slow down as it rises up the next hill, its inertia keeps it moving foward. Whenever you accelerate, you feel a gravity-like sensation "pulling" you in the direction opposite your acceleration. What you feel isn't really a force but it is your own inertia trying to keep you going in a straight line at a constant speed. This "pull" of inertia is sometimes called a "fictitious force."
An important aspect of the physics of a roller coaster is its acceleration. In a roller coaster, the cars are pulled up by a chain to the top of the highest hill along the track. Released from the chain as the front car begins its descent, the cars have almost no speed and only a small acceleration. As more cars get onto the downward slope the acceleration increases. The acceleration peaks when all of the cars are headed downward. As a steeper descent generates a greater acceleration, packing the cars with heavier passengers does not. When the coaster reaches the bottom of the valley and starts up the next hill, there is an instant when the cars are symmetrically distributed which gives it a zero acceleration.
Another important principle of roller coasters is their constant transition between energy states. When the chain hauls the cars to the top of the first hill, it does work on the cars thus endowing them with potential energy. As the cars descend into the first valley, much of the stored energy is turned into kinetic energy, the energy of motion. The roller coaster gains kinetic energy and speed at the expense of potential energy. Since kinetic energy is related to speed they both increase together: kinetic energy equals one half times the mass times the velocity squared. The roller coaster reaches maximum speed at the bottom of the first hill when all of its gravitational potential energy has been converted to kinetic energy. The roller coaster then rushes up the second hill, slowing down and converting some of its kinetic energy back into gravitational potential energy. The conversion of energy back and forth between the two forms continues, but energy is gradually lost to friction and air resistance so that the ride becomes less and less intense until it finally comes to a stop.
Here is a close examination of the constant interchange of energy.
1. The station brakes 'A' are released to allow the train to coast down gently sloping track 'B' and to engage with the powered lift mechanism 'C-D'. This is a drop in potential energy from 'A' to 'B' so that the train acquires kinetic energy and sufficient momentum to reach and lock to the chain.
2. At 'D' the train is at the highest point of the ride so it has acquired its maximum potential energy from the input of mechanical energy expended by the lift mechanism.
3. As the center of gravity of the train passes the top of the lift hill 'D' the train detaches from the lift mechanism and is free to accelerate down the first drop 'D-E'.
4. The train falls through a height (2) down the ramp'D-E' so the resultant potential energy drop from 'D' to 'E' is converted into kinetic energy which accelerates the train.
5. At the bottom of the first drop 'E' the track curves upwards so the riders and train will experience an upwards centripetal acceleration due to the change in velocity and thus the riders will feel heavier than normal.
6. The train rises again to hill crest 'F' exchanging potential energy gain for loss of some of the kinetic energy acquired by fall 'D-E' and losing speed in the process.
7. The process repeats itself as the riders crest hill 'F' then descend to dip 'G'. An impotant thing to note is that the bottom of the first drop 'E' is not the fastest drop of this particular ride. In this example not only is the distance 'F-G' greater than 'D-E' but the speed at 'G' can be calculated from the potential energy drop (5) which is clearly greater than drop (2).
8. After the dip 'G' and the upward rise 'H' the train reaches the 180 degree curved turnaround 'I'. This section must be well banked to avoid excessive side forces on the track. The speed of 'H' will be as the same as at 'E' because drop heights (6) and (2) are the same so the interchange of kinetic and potential energy between 'E-F', 'F-G' and 'G-H' balances out.
9. Point 'J' is the lowest point of the ride so 'J', is also, in theory, the fastest point of the ride. The greatest potential energy drop is 'D-J' which is described as the potential height (4) of the ride. The potential height is defined as the difference between the highest track surface and the lowest track surface for the entire circuit. This represents the maximal gravitational potential energy drop for the entire coaster and it is an indicator of the maximum speed which mught be achieved. In practice frictional and other losses will already have used up some of the available energy resulting in lower speed.
10. The train continues through smaller hills 'K' and 'L' until it reaches brake runs at 'M' and station 'A'. The friction surfaces of these brakes stop the train. The train's residual kinetic energy is converted to heat in the brake linings.
One specific type of roller coaster is the Double Loop. The coaster on the Double Loop is held onto the track by a double set of wheels, one set on the top of the rails and the other set on the bottom. If circular motion is to be maintained, the net force must be towards the center of the circle. At the bottom of the loop, the passenger's weight vector is pointing downward and the seat is acting in an upward force. Since the push from the seat is greater than the person's weight, the net force points toward the center of the loop and begins the circular motion. At the top of the loop, the two forces (the weight and the force from the seat) point downward and combine in the net force that makes the rider continue in the circle. For the roller coaster to work, the first hill must be higher than the top of the loop by at least half of the radius of the loop.
The choice of seat on a roller coaster makes a difference in the ride. At the first descent, the front car starts down slowly because most of the roller coaster's energy is potential. The speed of the cars increases as an exponential function of time, so that the rear end starts down at a much higher speed than the front. Although the passengers in the front of the car get an unobstructed view of the descent, the passengers in the rear car have a stronger sense of being hurled over the edge. To a passenger in the rear who is loosely held in place by a safety bar, a fast trip over a hill provides a brief sensation of being lifted from the seat and he arrives at the crest with a large momentum. Until he encounters the safety belt and is redirected, he continues to travel upward even thought the coaster has leveled out below. The faster the roller coaster goes over the hill, the greater the sensation of being hurled. Now that you know this information, where are you going to sit?
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References
Cutnell, John, and Kenneth Johnson. Physics: Volume I. New York: John Wiley and Sons Inc, 1998.
The Gravity Machine-Coaster Dynamics
Walker, Jearl. "The Amateur Scientist." Scientific-American. Oct 1983. v249. n4. pp.162-164,167-169.
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