How do the basic laws of physics allow a roller coaster to accelerate?
Gravity applies a constant downward force on the cars. The coaster tracks serve to channel this force — they control the way the coaster cars fall. If the tracks slope down, gravity pulls the front of the car toward the ground, so it accelerates.
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Most roller coasters run by the Law of Inertia. Since an object at rest stays at rest, all roller coasters have to be pushed or pulled to get started.
The force of gravity pulling a roller coaster down hill causes the roller coaster to go faster and faster, it is accelerating. The force of gravity causes a roller coaster to go slower and slower when it climbs a hill, the roller coaster is decelerating or going slower.
As the train enters the loop, it has maximum kinetic energy — that is, it is moving at top speed. At the top of the loop, gravity has slowed the train down somewhat, so it has more potential energy and less kinetic energy — it is moving at reduced speed. Originally, roller-coaster designers made circle-shaped loops.
The two most important forms for amusement park rides are kinetic energy and potential energy. In the absence of external forces such as air resistance and friction (two of many), the total amount of an object's energy remains constant.
The acceleration along the track is always equal for every car, but for each car that acceleration aligns with the hills/gravity in different ways. As the front car crests a hill, the coaster is decelerating; the front car is being pulled backward by the other cars.
Roller coasters, like everything else, must obey the law of conservation of energy, meaning the train can only go as fast and as far as the amount of stored (potential) energy allows. Potential energy usually comes from lifting the train up a hill with a chain or cable.