Math is used to calculate the height of the roller coaster and to produce the optimal results for speed after the roller coaster has left the pulley. Physics is the main element for designing roller coasters, but geometry is also used.
Roller coaster engineers work hard to develop new roller coasters. They first begin with an idea of what they want the latest and greatest roller coaster to be. They then put this idea onto paper. The engineers must calculate the perfect angles, heights and directions that will allow the roller coaster to reach great speeds without going off of the track. They use many machines, test coasters and models to come up with the perfect angles.
Engineers must first be sure that the angles and measurements work in the right way before they can go back to designing the actual roller coaster. They can come up with new ideas and extras to add to the coasters that will allow them to go faster.
Roller coaster cars do not have any engines. They are completely controlled by the angles and the velocity they reach while going down the first big drop. They are propelled by the power of gravity from that drop, and the parts of the roller coaster that do most of the work are the pulleys that get the coaster to the top of the first big hill and the mechanisms that help stop the roller coaster when the ride is over.