The Math of Roller Coasters

Reposted from

Designing a good roller-coaster loop is a balancing act. The coaster will naturally slow down as it rises, so it has to enter the loop fast enough to make it up and over the top. The curving track creates a centripetal force, causing the cars to accelerate toward the center of the loop, while momentum sweeps them forward. Loose objects like riders are pinned safely to their seats. The acceleration gives the ride its visceral thrill, but it also puts stress on the fragile human body—and the greater the velocity, the greater the centripetal acceleration.

Coney Island’s Flip-Flap Railway, built in 1895, reached a neck-snapping 12 times the force of gravity at the bottom of its loop—more than enough to induce what pilots call G-LOC, or gravity-induced loss of consciousness. In other words, riders often passed out. In fact, any vehicle trying to get around a perfectly circular loop has to hit at least 6 g’s—enough to render most people unconscious.

To solve the problem, modern designers adopted an upside-down teardrop shape called a clothoid, in which the track curves more sharply up top than at the bottom. Then most of the turn happens at the peak, when the coaster is moving the slowest and the acceleration is least. Result: no G-LOC, just screams. The formula that helped them do it? ac = v2⁄ r.

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