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Modeling Roller Coaster Functions: A Mathematical Approach

keiran avatar
keiran
12 diapositives · 281 Views · Aug 13, 2014

This project explores the mathematics behind amusement park rides, specifically roller coasters. We analyze the height of riders using algebraic functions, determining ride durations and heights at different points. Through the example rides, we derive equations representing their paths, examine maximum heights and durations, and design a roller coaster using piecewise functions. By applying mathematical principles, we gain insights into ride dynamics and safety, enabling better design and engineering decisions for amusement attractions.

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Modeling Roller Coaster Functions: A Mathematical Approach
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  1. Math Project Done by :Ahmed Abdullah & KalifaRashid Grade: 12.04

  2. Roller Coaster Fun

  3. TASK 1: Equation Modeling • Below are descriptions of some rides in an amusement park and some questions for you to answer. Use algebra (i.e. do not use your graphing calculator to graph the functions) to solve each unless otherwise directed. SHOW ALL YOUR WORK • 1. The SLIDE. This ride is located in the kiddie section! The height of the rider above ground, h yards, after t seconds can be modeled by the function: • A. How long does the ride last (from starting height to reaching ground level)?  h(t) = -0.5t + 40  h(t) = 0 , So 0.5 t = 40  t = 80 seconds • B. Change the numbers so that the ride starts higher and drops faster • if ht= -2t+80  then ride starts higher and drops faster • c. Now how long does the ride last, based on the changes in part b? ht= -2t+80  So t= 40 seconds

  4. Question 2: The LITTLE DROP • The LITTLE DROP. On this ride, for some period of time, the rider dips below the ground level. The height of the rider after t seconds can be modeled by the function: • What is the starting height of this ride? h(t) = 4t2 +44t+96  h(0) = 96  starting height = 96 • How long is the rider below the ground? h(t) = 0  4t2 +44t+96 = 0 (t-8) (t-3) = 0  t=8 or t=3 • If the ride lasts a total of 10 seconds, what is the height of the exit gate? h(10) = 400-440+96  h(10)= 56  So the height of exit gate = 56

  5. Use the information from the parts above to sketch a graph of the height of the ride over time with the appropriate labels on the axes. 96

  6. Question 3: The SCREAM This ride lasts for 8 seconds. The height of the rider can be modeled by the function: h(t)=(-6t2+12t)-(t2-12t+32) At which height does this ride begin? t=0  h(0)=(-6(02 )+ 12t ) ( 02-12(0)+32 ) h(0)=32m At what height does this ride end? • t=8→h(8)=(-6(82 )+ 12t ) ( 82-12(8)+32 )  h(8)=0m At what height is the ride after 5 seconds? t=5  h(5)=(-6(5)2 + 12t ) ( 52-12(5)+32 )  h(0)=32m • At what time(s) does the ride hit ground level?

  7. At what time(s) does the ride hit ground level? ht=0 when t=?? ht=(-6t2+12t)(t2-12t+32) ht=-6t4+72t3-192t2+12t3-144t2+384t ht=-6t4+84t3-336t2+384t 0=-6t4+84t3-336t2+384t -6t4-14t3+56t2-64t=0 -6tt3-14t2+56t-64=0 -6tt-2t-4t-8=0 t=0 t=2 t=4 t=8 Therefore, the ride hit ground levelwhen t=0,2,4 and 8

  8. Sketch a graph of the ride with the appropriate labels on the axes. You may use you’re the Grapher program and experiment with the WINDOW to get the right picture.(copy paste your graph below)

  9. Task 2: The Design • A roller coaster can be based on mathematical functions, but they are more likely to be made up of pieces, each of which is a different mathematical function. This allows much more flexibility. Use piecewise functions to design your own roller coaster. (You will actually be modeling the height of the roller coaster over time.) Include a graph of the roller coaster. Your roller coaster MUST fit the following criteria: • a. Each of the pieces are connected to each other • b. Uses at least 5 DIFFERENT functions/pieces • Once you have decided on a design of your roller coaster and have graphed it answer the following questions using the graph and the function.

  10. a. What is its starting height? 2 Meters b. How long does the ride last? 98.3 seconds  ( 1 minute 40 second ) c. What is its ending height? 0 meter d. Approximate how long it takes to reach its highest point? 82 Seconds e. Why would it be useful to be able to have equations for a roller coaster? It’s useful because we can now the exact time to reach the maximum height and we can calculate the velocity of every point. f. What kinds of things might you be able to figure out about the roller coaster? We can calculate the acceleration, Velocity, Slope and gravity those things should be calculated to know that Roller coaster will not be pulled by the gravity force and it will run in smooth way.

  11. g. How might it help you to design or change the design? As I have mentioned the help to design a smooth and sharp way to the roller coaster, the design should have as mush safety as possible to the rider. The design can be changed by changing the formulas making it wider, higher and sharper.