Speed at the top of an elliptical roller coaster loop

In summary: Apparently the shape they use is called a clothoid. Its designed to reduce the Max g passengers experience.
  • #1
JessicaHelena
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Homework Statement


This isn't really a homework but a project I'm doing that's due soon. In our multivariable calculus class, we're creating a 3D roller coaster, and I need to explain the physics behind the roller coasters.

For a roller coaster loop, if it were perfectly circular, we would have a minimum speed of ##v_{min} = \sqrt{gR}## at the top of the loop where ##g=9.8 m/s^2## and ##R## is the radius of the 'circle'. However, most roller coaster loops are actually not circular but more elliptical. I've been looking for ways to calculate the min. speed at the top for an elliptical loop, but so far I haven't been able to. How could I go about that?

Homework Equations


F_net = ma_c

The Attempt at a Solution


I really didn't know how to do this, so I searched google, but as far as I can see, there aren't any explanations for this...
 
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  • #2
An ellipse is not a circle, but it will have a radius of curvature associated with each point of the curve. You could use the radius of curvature at the top as you would the radius of a circle. Your task will be to find the radius of curvature for the ends of the ellipse.

Start by googling "ellipse radius of curvature".
 
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  • #3
gneill said:
An ellipse is not a circle, but it will have a radius of curvature associated with each point of the curve. You could use the radius of curvature at the top as you would the radius of a circle. Your task will be to find the radius of curvature for the ends of the ellipse.

Start by googling "ellipse radius of curvature".
Right, but it raises the possibility that the top of the loop is not the only concern.
 
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  • #4
haruspex said:
Right, but it raises the possibility that the top of the loop is not the only concern.
Indeed it does. We also don't know how the ellipse is oriented: major axis vertical? Horizontal? Something else?. I've been assuming vertical.

The curvature can be found for any point along the curve, so it might be worth checking. The curvature at the ends of the major axis is a very simple expression as it turns out (also for the ends of the minor axis). I expect it to be much more hairy elsewhere. Finding a general expression for the track normal force around the ellipse might be doable. But I don't think I'm going to do it :smile:
 
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Related to Speed at the top of an elliptical roller coaster loop

1. How fast do roller coasters go at the top of an elliptical loop?

The speed of a roller coaster at the top of an elliptical loop varies depending on the design of the coaster and the forces acting on it. However, most roller coasters reach speeds of 30-50 miles per hour at the top of an elliptical loop.

2. What affects the speed of a roller coaster at the top of an elliptical loop?

The speed of a roller coaster at the top of an elliptical loop is affected by several factors including the initial launch speed, the design of the loop, and the forces of gravity and friction. The shape and size of the loop also play a role in determining the speed at the top.

3. How is the speed of a roller coaster at the top of an elliptical loop calculated?

The speed of a roller coaster at the top of an elliptical loop can be calculated using Newton's second law of motion, which states that force equals mass multiplied by acceleration. The acceleration of the coaster can be determined by dividing the change in velocity by the time it takes to complete the loop.

4. Why do roller coasters slow down at the top of an elliptical loop?

Roller coasters slow down at the top of an elliptical loop due to the forces of gravity and friction. As the coaster travels up the incline of the loop, its speed decreases due to the force of gravity pulling it back down. Friction between the coaster and the track also acts to slow down the speed at the top.

5. Can the speed of a roller coaster at the top of an elliptical loop be increased?

Yes, the speed of a roller coaster at the top of an elliptical loop can be increased by adjusting the design of the coaster, such as increasing the initial launch speed or decreasing the forces of friction. However, safety regulations and the physical limitations of the human body must also be taken into consideration when designing a roller coaster loop.

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