When the roller coaster moves downwards, kinetic energy is generated. The maximum kinetic energy generated is when the roller coaster is at the bottom of the track. When it begins to go up, the kinetic energy converts to potential energy.
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The roller coaster cars gain potential energy as they are pulled to the top of the first hill. As the cars descend the potential energy is converted to kinetic energy. The coaster cars have the maximum kinetic energy they will ever have throughout the ride.
At the bottom of the first hill, your kinetic energy is at its highest point. You're going as fast as you'll ever go on this roller coaster ride. To ensure the fun keeps going, the roller coaster's designers put in the second hill. If the first hill were the ride's only one, the fun would be over sooner.
Kinetic energy - the energy of motion - is dependent upon the mass of the object and the speed of the object. The train of coaster cars speeds up as they lose height. Thus, their original potential energy (due to their large height) is transformed into kinetic energy (revealed by their high speeds).
Basic mathematical subjects such as calculus help determine the height needed to allow the car to get up the next hill, the maximum speed, and the angles of ascent and descent. These calculations also help make sure that the roller coaster is safe. No doubt about it--math keeps you on track.
In roller coasters, the two forms of energy that are most important are gravitational potential energy and kinetic energy. Gravitational potential energy is the energy that an object has because of its height and is equal to the object's mass multiplied by its height multiplied by the gravitational constant (PE = mgh).