Question

Now, make one final coaster that has the works! Include each of the following: a. A "swoop" down through the x-axis at x = 300. (This should follow an initial rise of the track.) b. A simple crossing upward through the x-axis at x = 700. c. A smooth ending at x = 1000 using an exponent of 2.

Answer 1

Roller coaster design incorporates physics principles such as force, energy, and centripetal acceleration. Track components can be designed using trigonometric and parabolic functions. The kinetic energy of a cart varies due to potential energy conversion throughout the ride.

Step-by-step explanation:

Understanding Roller Coaster Physics

Designing the ultimate coaster involves understanding key principles of physics and applying mathematical functions to create various track sections. Let's consider these sections as functions of position x along the track:

A 'swoop' down through the x-axis at x = 300 could be modeled by a negative cosine function with a phase shift, reflecting the initial rise followed by a sharp decrease.

An upward crossing through the x-axis at x = 700 could be a positive zero crossing of the sine or cosine function, signifying a gentler slope.

A smooth ending at x = 1000 with an exponent of 2 suggests the use of a parabolic function, like y = k(x - 1000)^2, where k is a positive constant to ensure the curve opens upward.

Considering the kinetic energy of a cart on a roller coaster track modeled by a perfect cosine function through its first period, we can expect fluctuations. The cart will have maximum kinetic energy at the bottom of the track where potential energy is at its least, and minimum kinetic energy at the peak where potential energy is highest, given conservation of energy in a frictionless system.

In a real-world scenario with a track featuring a vertical loop, the speed at the top of the loop is crucial for safety. If the radius of curvature is 15.0 m and the desired downward centripetal acceleration is 1.50 g (where g is the acceleration due to gravity), we must calculate the velocity using the equation for centripetal acceleration, a = v^2/r, combining this with gravitational force to meet the safety requirement.

Answer 2

Adjusting the coefficients and functions allows to fine-tune the shape of the coaster track. The quadratic and exponential terms contribute to the smoothness of the transitions, while the sinusoidal and logistic terms add dynamic and interesting features to the coaster's path.

Creating a coaster track with the specified features involves defining a mathematical function that represents the coaster's height as a function of the horizontal position (x). Here's a possible function that incorporates the requested features:

`\begin{aligned}& h(x)=(1)/(100)(x-300)^2 \cdot \sin \left((2 \pi(x-300))/(100)\right)+150+(100)/(1+e^(0.02(x-700)))+20 . \\& e^(-0.005(x-1000)^2)\end{aligned}`

Swoop Downward at x=300:

The term
`(1)/(100)(x-300)^2` represents a quadratic function that makes the coaster swoop downward.

`\text { The } \sin \left((2 \pi(x-300))/(100)\right)` part adds oscillations to create a smooth and interesting curve as the coaster descends.

Initial Rise of the Track:

The constant term 150 is added to lift the entire coaster track, providing an initial rise.

Crossing Upward at x=700:

The term
`(100)/(1+e^(0.02(x-700)))` represents a logistic function. As x approaches 700, the function approaches 100, causing the coaster to rise smoothly.

Smooth Ending at x=1000:

The term
`20 \cdot e^(-0.005(x-1000)^2)` represents a Gaussian or bell-shaped curve. As x approaches 1000, the coaster smoothly descends.

You can use this function to calculate the height of the coaster at any given x position. Adjust the coefficients or add more terms based on your preferences for the coaster's shape.