The goals of this project are: To define the piecewise function formulas defining the coaster To create a replicated graph of the formulas To determine the maximum height of the coaster as well as the longest drop, in comparison to other great coasters in the world To determine how long the passengers will be in the air, how fast the coaster is going when hitting the “splashy zone”, and where the “splashy zone” should be built To determine the maximum height a passenger can reach during this ride, and how long it takes to reach that height.
The values of the unknown constants in the piecewise function can be known by understanding that everywhere on the coaster is smooth and seamless, without gaps or humps. Therefore, at points where the functions change, we can apply continuity (the coaster rails are smooth) and differentiability (no sharp turns on the rails) to derive the two equations needed to find the value of the constant.
To ensure that the roller coaster is continuous at x=5,
Ax^2-1085=972x+D
Since at x=5, both functions share the same instantaneous velocity,
ⅆ/ⅆx [Ax^2-1085]=ⅆ/ⅆx[972x+D]
2Ax=972
10A=972
A=97.2
By substituting A=97.2,
97.2x^2-1085=972x+D
At x=5,
97.2〖(5)〗^2-1085=972(5)+D
D=-3515
To ensure that the roller coaster is continuous at x=5 with knowing the value of D,
972x-3515=Ex^3+Fx^2+8100x-66155 …show more content…
If we know the time required for the coaster to reach maximum height during flight, we can derive the maximum height of the coaster’s flight and compare that to the maximum height of the coaster, then finally determine the highest position a person can get during this ride. Since the coaster will stop moving when it reaches maximum height while flying, its velocity will become 0 at that point in
The values of the unknown constants in the piecewise function can be known by understanding that everywhere on the coaster is smooth and seamless, without gaps or humps. Therefore, at points where the functions change, we can apply continuity (the coaster rails are smooth) and differentiability (no sharp turns on the rails) to derive the two equations needed to find the value of the constant.
To ensure that the roller coaster is continuous at x=5,
Ax^2-1085=972x+D
Since at x=5, both functions share the same instantaneous velocity,
ⅆ/ⅆx [Ax^2-1085]=ⅆ/ⅆx[972x+D]
2Ax=972
10A=972
A=97.2
By substituting A=97.2,
97.2x^2-1085=972x+D
At x=5,
97.2〖(5)〗^2-1085=972(5)+D
D=-3515
To ensure that the roller coaster is continuous at x=5 with knowing the value of D,
972x-3515=Ex^3+Fx^2+8100x-66155 …show more content…
If we know the time required for the coaster to reach maximum height during flight, we can derive the maximum height of the coaster’s flight and compare that to the maximum height of the coaster, then finally determine the highest position a person can get during this ride. Since the coaster will stop moving when it reaches maximum height while flying, its velocity will become 0 at that point in