Coaster Dynamics: Coaster Lab #3



Coaster Dynamics
Coaster Lab

Lesson #3:   Newton's First Law of Motion:   Free-Falling for You

TOPICS:   Newton's First Law of Motion and Projectile Motion.

1.   Starting with the default track configuration in Coaster Dynamics, change the Round Loop of Module 2 to a Single Hill, and set the height of the first 2 elements to:

        Spiral Hill = 30 m
        Single Hill = 30 m.

2.   Run the roller coaster ride and stop it after the car passes the hills. For the Single Hill, use the Coaster Lab feature to determine the vertical G force at Watchpoints 6-8 (ie., the start of the downhill portion of the hill). What is the veloctity, vx, at the top of the hill (ie., at Watchpoint 6)?


6 7 8
Gy
vx

3.   For maximum thrills, one goal of roller coaster designers is to create hills which will simulate the feeling of free fall (ie., Gy = -1.0) for as long as possible. In Coaster Dynamics, the shape of all the hills are sinusoids (this makes the math easier to solve, but the hills less exciting). Instead, to simulate free fall, a hill could be designed so that its shape, as you drop over the top, is the same curve that would happen if the car were in true free fall. In this case, the vertical force would remain -1.0 G, and the rider would feel weightless.

In Chapter 7 of the Physics Primer, it is shown that the trajectory for free fall is a parabola. Unfortunately, a simple parabola can't be used as a hill since it never changes direction upward -- meaning you would eventually crash into the ground. One possible solution to this problem is to design a hill that combines a parabola with a circular arc, as illustrated in the figure below.

Figure 1. Geometry of a Simple Free Fall Hill.

In this hill, the top portion is a parabola, and at some point before the bottom of the hill, it then changes into a circular arc of 45 degrees, and of radius, R.

From Chapter 7 of the Physics Primer, the equation for a free fall parabola is:

(1)         x² = (-2vx²/g)(y).

At the transition point, P, a little trigonometry will show that for an arc of 45 degrees, the y-coordinate is equal to:

(2)         Py = ( 1 - cos(ß) )(R) - H

where ß = 45 degrees.

To simplify the solution, we will also make the assumption that the transition point, P, is at the vertical midpoint of the hill. That is, the y coordinate of the transition point is equal to:

(3)         Py = -H/2.



Using equations (1), (2) and (3), calculate the coordinates of the transition point, Px and Py, the value of the arc radius, R, and the coordinates of the center of the arc, Cx and Cy, using a total hill height of H = 30 m, and the velocity at the top of the hill, vx, you found in Part 2 above.































NOTE:   Although this type of hill is similar to those in real roller coasters, the true design is more complex than in this analysis. The problem with this hill is that at the transition point, the slopes of the curves are different, and as a result, there would be an abrupt change in the G force -- which would be jarring to the riders. Instead, in real roller coasters, the hills must be designed to have the same slope at the transition point, which will make for a smoother ride. For those desiring a challenge, you can design a Free Fall Hill with a smooth transistion point in the Advanced Study section that follows.



ADVANCED STUDY (optional)

In order to design a Free Fall Hill with a smooth transition point between the parabola and arc, some calculus will be helpful.

To achieve geometric continuity at the transition point, we must remove the constraint that the transition point occurs at the vertical midpoint of the hill. Thus, we allow the position of the transition point to be a variable, and instead, the constraint we now impose is that the instantaneous slopes of the parabola and arc must be equal at the transition point.

To help guide you in finding a solution, try to follow the steps below.

(A)   The instantaneous slope of a curve is equal to the dervative of the equation, evaluated at the point of interest. The curve we want to analyze is the position, y versus x, for the parabola. So, first derive an equation for the derivative, dy/dx.












(B)   For the arc at the bottom of the hill, the arc extends through an angle of 45 degrees at the transition point, P. For a circle at 45 degrees, the slope is equal to -1.0 (negative because y decreases as x increases). We want the parabola at point P to have the same slope. Therefore, set the equation you derived above for dy/dx equal to -1.0, and solve for x.












(C)   Using the value of vx you found in Part 2 for the speed at the top of the hill, solve for x -- which equals Px, the x-coordinate of the transition point.











(D)   Using equation (1) in Part 3 above, solve for the y coordinate of the transition point, Py.











(E)   Using equation (2) in Part 3 above, and a total hill height of H = 30 m, solve for the arc radius, R. And finally, calculate the x and y coordinates for the center of the arc, Cx and Cy.













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