# A Wild Ride - roller coaster physics

• rsala
In summary, the roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding 20m/s. The maximum value of A allowed by these regulations is 1.66183.
rsala

## Homework Statement

A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x-axis be parallel to the ground and the positive y-axis point upward. In the time interval from t=0 to t=4 s, the trajectory of the car along a certain section of the track is given by
http://img526.imageshack.us/img526/719/renderxg2.gif

where A is a positive dimensionless constant.

The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding 20m/s. Find the maximum value of A allowed by these regulations.

## Homework Equations

A = $$\sqrt{a_{x}^{2} + a_{y}^{2}}$$

## The Attempt at a Solution

well, I have separated this equation into 2 components of position, rx ry
Rx = A(t)
Ry= A(T$$^{3}$$ - 6T$$^{2}$$)

took the derivative of each component to change R to V

Vx = A (this is A because, A is a constant and i just treated this as i took the deriative of any constant next to a variable with power of 1, just kept the constant.)
Vy = A(3T$$^{2}$$ - 12T)

The magnitude of this vector V is
V = $$\sqrt{ A^{2} + (3T^{2}-A12T)^{2}}$$now.. my problem here is how can i find which maximum value of A whose speed doesn't pass 20, i HAVE thought of setting this equation to 20, but what about -20 velocity, since it asks for speed not velocity...this rollercoaster CAN go downward.

Last edited by a moderator:
Okay, so what is that you are supposed to do again? You are right, except your velocity isn't right (but your idea is!), so far as I can tell what you are supposed to do.

im supposed to find what highest value of A can be put in so the rollercoaster doesn't speed past 20 m/s,

homework was due , couldn't get it, answer was 1.666 ...how can you get this?

Ah, okay. Whether or 20 is negative or positive would be irrelevant for the magnitude of the velocity.

So let's continue with what you did:

$$V = \sqrt{v_x^2 + v_y^2}$$

$$V = \sqrt{A^2+A^2(3t^2-12t)^2}$$

$$A = \frac{V}{\sqrt{1+(3t^2-12t)^2}}$$

So now we see that the amplitude will depend on time because the velocity is defined to be less than 20m/s. We also want to maximum, so let's take the first derivative of A with respect to time.

$$\frac{dA}{dt} = 0 = -\frac{1}{2} \frac{V}{\sqrt{1+(3t^2-12t)^2}}*2(3t^2-12t)(6t-12)$$

This equation looks sort of beefy, and because I have done enough algebra to last a lifetime I put it into mathematica.

Mathematica, a math program, tells me (after plugging in 20 for V) that the time derivative is zero, i.e. the maxima occur at 0,2,4. The zero and 4 times are out of our control, and we don't really care about them. So let's go back to A(t) and plug in 2 for t.

A(2) = 1.661

## 1. How does a roller coaster stay on the track?

A roller coaster stays on the track through a combination of gravity and centripetal force. The initial drop of the coaster creates potential energy, and as it goes through loops and turns, the centripetal force pulls the train towards the center of the track, keeping it from flying off.

## 2. What is the highest point of a roller coaster called?

The highest point of a roller coaster is called the apex or crest. This is usually the point where the coaster reaches its maximum potential energy before beginning its descent.

## 3. How does the height of a roller coaster affect its speed?

The higher a roller coaster is, the more potential energy it has. This potential energy is then converted into kinetic energy as the coaster descends, increasing its speed. Therefore, a higher coaster will typically result in a faster ride.

## 4. How do roller coasters use physics to create thrilling sensations?

Roller coasters use a variety of physics principles to create thrilling sensations. For example, sudden drops and twists use gravity and centripetal force to create the feeling of weightlessness and acceleration. Inversions use the forces of acceleration and deceleration to create the feeling of being upside down.

## 5. How do engineers design safe roller coasters?

Engineers use various safety measures to design safe roller coasters, including computer simulations, physical tests, and strict regulations. They also consider factors such as forces on the body, structural integrity, and emergency systems to ensure the safety of riders. Additionally, regular maintenance and inspections are crucial to ensure the continued safety of the ride.

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