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Roller Coaster Physics
Gravity and Potential Energy
Gravity is the driving force of a roller coaster. From the moment the roller coaster train
passes the peak of the lift hill, it is the acceleration due to gravity that brings it back to the
beginning. When the train is released from the top of the lift hill, gravity pulls it down.
The train begins slowly, then picks up speed as it approaches the bottom of the hill. As it
begins to climb the next hill, the speed decreases. This is the acceleration due to gravity,
and it occurs at 9.80m/s2 straight down toward the center of the Earth. The initial hill, or
the lift hill, is the highest in the entire ride. As the train is pulled to the top, it is gaining
potential, or stored energy. The higher the lift, the greater the amount of potential energy
that is gained by the train. This is shown by the equation for potential energy:
Ug = mgh
Where Ug is potential energy, m is mass in kilograms, g is acceleration due to gravity
(9.80m/s2), and h is the distance above the ground in meters. Because mass and gravity
are constant for the train, if the height of the train above the ground is increased, the
potential energy must also increase. This means that the potential energy for the roller
coaster system is greatest at the highest point on the track: the top of the lift hill.
Velocity and Kinetic Energy
As the roller coaster train begins its descent from the lift hill, its velocity increases. This
causes the train to gain kinetic energy, which is the energy of motion. The faster the train
moves, the more kinetic energy that is gained by the train. This is shown by the equation
for kinetic energy:
K = 1/2mv2
Where K is kinetic energy, m is mass in kilograms, and v is velocity in meters per
second. Because the mass is constant, if the velocity is increased then the kinetic energy
must also increase. This means that the kinetic energy for the roller coaster system is
greatest at the bottom of the highest hill on the track: the bottom of the lift hill.
Conservation of Energy
Energy cannot be created or destroyed, but it can be converted from one form to another.
For the idealized roller coaster, all energy is conserved through conservative forces, i.e.
gravity. As the train accelerates down the lift hill, potential energy is converted into
kinetic energy. When the train ascends another hill, the kinetic energy is converted back
into potential energy again. This is conservation of mechanical energy, and it continues
throughout the entire ride. The total mechanical energy for the train is shown by the
equation:
E =K + U
Where E is the total mechanical energy, K is kinetic energy, and U is potential energy.
From this, the equation for conservation of total mechanical energy can be derived:
Ei = Ef
Ki +
or
Ui = K f
+ Uf
Where Ei is total initial mechanical energy and Ef is total final mechanical energy. It is
because of this phenomenon that a roller coaster is called a “coaster.” After the initial
input of energy to carry the train up the lift hill, the roller coaster simply coasts through
the rest of the ride.
Friction
Not all of the energy in the roller coaster system is conserved in a non-idealized system.
This is because of nonconservative forces. Friction is the main cause of energy leaks in
the system and the reason why mechanical energy is not fully conserved. The friction
between the train and it’s tracks as well as between the train and the air take energy out of
the system, slowing the train and creating both heat and sound. This effect is most
noticeable at the end of the ride as all enduring kinetic energy is taken out of the system
though brakes. Because of the energy leaks due to friction, each successive hill or loop
on a roller coaster must be shorter than all the hills or loops previous to it, otherwise the
train will not make it all the way up.
Centripetal Acceleration
Curves are an essential part of a roller coaster, and centripetal acceleration is part of
moving in a circular path, so this acceleration is also an essential part of a roller coaster.
Centripetal acceleration points toward the center of the circular path of the train, but is
felt by passengers as a force pushing them to the outer edge of the circular path. This
feeling is often described as centrifugal force, although there is no actual force pushing or
pulling passengers away from the circle. The “centrifugal force” is actually your body’s
resistance to the train’s change in direction: your body wants to continue in a straight line
and attempts to do so as the train turns. The equation for centripetal acceleration is:
at = v2 / r
Where at is centripetal acceleration, v is velocity in meters per second, and r is the radius
of the circle in meters. This means that the higher the train’s velocity, the greater the
centripetal acceleration. This also means that the smaller the curve of the path being
traveled, the greater the centripetal acceleration. Because of this, many high-speed roller
coasters use banked turns rather than the flat ones that are safe for slower speeds.
Banking the turns in a roller coaster gives you the feeling of being pushed into your seat
rather than being thrown to the side of the car.
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