Have you ever ridden a roller coaster? The people who design these rides need a deep understanding of math and physics. They have to calculate how big to make the hills, how fast the roller coaster will move at various points on the track, and how long the ride should last. The equation at the very heart of all these calculations is a quadratic equation.

#### One Equation to Rule Them All

When a roller coaster designer needs to make calculations, she starts with a very basic formula:

\begin{align*}\text{distance}=\text{(initial velocity)(time)}+\frac{1}{2}\text{(acceleration)(time)}^2\end{align*}

This is a quadratic equation, with time as the variable. The equation allows her to figure out how far the roller coaster train can travel once it starts down the first hill. Using this equation, the designer can also calculate when the coaster will hit its top speed and what that speed will be. It provides a starting point for understanding how the coaster will behave on climbs and at turns, and whether it can arrive safely back at the station.

Designers have to make sure their coasters are safe long before construction begins. Coasters are expensive to build, so designers have to verify that their proposed ride will balance safety and thrills before it's made into reality. Yet, even with all of these precautions, roller coasters still carry some risk.

See for yourself: http://www.nbcnews.com/video/nightly-news/52538877#52538877

#### Explore More

Check out the links below to learn more about how roller coasters use math and physics to deliver a thrill.

http://www.nytimes.com/2012/07/03/science/wood-takes-a-thrilling-turn-in-roller-coaster-design.html

http://www.youtube.com/watch?v=cao4JSfaA7s

http://www.youtube.com/watch?v=K36iWnSaOEQ