**Calculus in Roller Coasters**

Introduction:

The Principles

Engineering in Roller Coasters

Safety Regulations

Project Description

~ The development and work-ability of a roller coaster, which involve the speed, acceleration, slope, and maximum and minimum height, are all found through the application of calculus.

~ Significance:

- The safety of the people riding these roller coasters.

~ Reason?

- To build a better roller coaster.

~Without the application of calculus, a roller coaster would not be able to complete it's course or operate as expected.

The number one priority of all amusement parks is safety.

Engineers make sure that everyone is safe at all times so the customers can experience the full satisfaction of riding a roller coaster.

They have to carefully plan out their blueprints or else people can seriously get hurt.

In order to prevent injuries, engineers have to make sure that the roller coaster meets all safety requirements.

http://en.wikipedia.org/wiki/Roller_coaster#Safety

Precision

When constructing roller coasters, Calculus is used by engineers to precisely calculate the curves, loops, and twists on a roller coaster

Prediction

Calculus, with the help of physics allows engineers to predict the speed of the roller coaster at a given height and slope.

This makes sure that the roller coaster speed is safe and will determine how fast it is going at a certain point

Engineers are faced with tasks such as finding the the velocity at the maximum height of the loop to make sure the g-force is safe for the riders

When engineers create hills and valleys, they have to take into account the acceleration of the roller coaster.

Calculus in Roller Coasters

Erwin Gomez

David Flores Ponce

Claire Ratliff

Oscar Zavala

The maximum height of a roller coaster is determined by using calculus.

The height of the roller coaster is called its "maximum."

the slope changes from positive to negative, creating a curve.

Both must be continuous and differentiable at x=0.

By:

The acceleration and velocity of the moving cart are also important to the operation of a roller coaster and are determined through the use of calculus.

Position

Derive to find velocity

Derive again to find acceleration

Conclusion

Calculus is used in the construction of roller coasters in the engineering field. In this project, we learned that calculus is used to calculate the maximum and minimum height of roller coasters and the velocity and acceleration the roller coaster needs to have to successfully operate in real life.

Calculus is also an important component in calculating the maximum and minimum heights the roller coasters can go.

How Calculus is Applied

h(t)=0.3t^3-5t^2+21t

h'(t)=v(t)=0.9t^2-10t+21

v(2.7)=.5610003 m/s

We found the velocity at the maximum and minimum.

PE=mgh

PE=26(10)(700)=182000

KE=(1/2)mv^2

PE=KE

182000=(1/2)(700)v^2

v= 22.8m/s

2.32 g's (g-force)

.561 m/s

22.8 m/s

0.3t^3-5t^2+21t

Calculus

Physics

http://www.mrwaynesclass.com/ap/coaster/web/index08.html