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Calculus and coasters


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My professor knows I like coasters, so when I asked him a question about coasters and Calculus that he couldn't answer, he said I could go find out for extra credit.


We're learning derivatives and we were discussing the applications of derivatives. Coasters came up when we were discussing the "jerk" function... I think the 3rd derivative of position, the 2nd derivative of velocity, the first derivative of acceleration.


Two questions that I'd really appreciate if anyone can help me answer!


1) What ways do derivatives assist roller coaster engineers?

2) (Something I've been wondering that my professor couldn't answer)... is there a reason Formula Rossa is a lot faster than Kingda Ka but not nearly as high? Is it because the forces the body would experience going nearly 150 mph and over 450 feet high are too extreme? Or is it merely a difference in the manufacturing of the two coasters? I'm wondering how possible it is from an engineering perspective for a coaster to be as tall/taller than KK and as fast/faster than Formula Rossa?


I'm not an Engineering major, I'm an English major, but I decided to take Calculus instead of Statistics partially so I could better understand and appreciate the math that goes into engineering coasters. If someone could give me a good explanation to this that doesn't mind me copying their answer for my Calculus teacher, I'd really appreciate it!

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For your first question jerk physics is used by roller coaster engineers to work out the maximum accelleration and maximum jerk a human body can take while also making the ride enjoyable.


Jerk is the rate of change of acceleration. Simply put if a roller coaster is designed to accellerate too fast over a short period of time most likely the riders will be, at least, uncomfortable and, at most, injured (due to whiplash for example).


As acceleration can be measured in both positive and negative values so that same consideration has to be applied to deceleration. So brake runs have to slow the rider gradually over a specific length of time to avoid uncomfort and injury.


Also of note is that even if the train is turning around a bend in the track while keeping the same speed, it is still tecncially accelerating as it is changing direction. So again a drastic change in direction at any speed will cause "jerk". So when designing a ride, for roller coaster engineers I'd imagine jerk physics would be extremely important in creating an enjoyable ride.




Oh and for your second question about Formula Rossa. Jerk physics comes into the equation if you are accelerating very rapidly and then try to go vertical rapidly. The body of the rider would experience significant forces if it wasn't done right. At the speed that coster reaches the accent to vertical would have to be a gradual one. In my opinion that wouldn't make for a very thrilling ride. Also I think Formula Rossa was built to mimic what it's like to be in a Formula 1 car. So plenty of speed, close to the ground so you feel that speed a bit more, as well as some tight turns.

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To clarify the jerk a little bit more:


The human body has limits to how much G's it can handle. But what also matters is how fast and how high those G's are applied.


If you would experience 4 G's for example, it would be OK, because your body can handle that. But it could be dangerous if

those 4 G's are applied from 0 G's within a split second. But if the force is applied gradually, it would be more comfortable.


Using a limit for the amount of jerk ( in G/s or m/s^3) can help you make a ride experience smoother.

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Thanks guys! I understand the jerk function a lot better now. Since we're mainly focusing on derivatives for this next exam, does someone have an example about how derivatives can apply in this situation? I've been understanding derivatives well enough but I'm still a little fuzzy about why they're important.


Stick with statistics

Too late. The semester is already halfway over and next semester I already have 18 units planned. I probably will never take math again in my life.

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well jerk is just the third derivative of position in respect to time.


So if you know the position, and therefore velocity over a time period you can work out the acceleration, and then work out the jerk.


I'm not sure I understand what you're asking. But a derivative is a measure of how a function

changes as its input changes. So for jerk, the third derivative, the input will be the change in acceleration. Using that derivative over a time period will give the rate of change of acceleration, which is the jerk. So really derivatives are just equations that make calculations a little easier.


Its not easy to explain simply but the way I consider it once you know the acceleration you dont need to know the velocity, as the acceleration comes from velocity over time. So in

any equation with velocity over time you just substitute one value, acceleration, which is the derivative of velocity. Once you know the acceleration over time you can calculate the jerk.


And as we mentioned before jerk is an important factor in designing comfortable roller coasters and to calculate it you need the third derivative of position in respect to time.


Where it would get complicated is that all points on the coaster track would have to have the jerk calculated to ensure a smooth ride, as at all points the velocity and acceleration, and therefore jerk, will be different.


There is a fourth derivative, measuring the rate of change of jerk, which could be used to do that.But such calculations are beyond my lowly electronic engineering brain!

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I would guess that the layouts of most modern coasters are designed with 6th order equations, so the jerk is defined by a cubic function. With these high-order curves, even the first and second derivatives of the jerk do not have any discontinuities.

In my cam design class we used up to 8th order curves for some of the cam profiles.

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