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# Calculus in Roller Coasters

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## Oscar Zavala

on 16 May 2014

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#### Transcript of Calculus in Roller Coasters

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
Full transcript