Acceleration through a non-uniform curve

[xoon]

So we are designing a roller coaster for a project, the roller coaster starts at 100m above ground and it has to go through all of these elements. for the whole ride, i need to find the lateral and the up/down g forces that are acting on the rider.

for the roller coaster itself, i have made a function for the roller coaster but i do not know how to get the acceleration components.

what i have so far is that v=ds/dt. but the problem is that my velocity does not depend on time, it depends on height because of conservation of energy. my velocity right now is.. v=Sqrt(2*g*(ho-h)). i do not know where to go from here, because it does not depend on time. i need acceleration in terms of position.

some other equations I've looked at are dv/ds*v=a(s) --> v*dv=a(s)*ds

i've been staring at this for about a day now and the project is due monday. please help..

 

[Bob S]

Also look at Isaac Newton's brachistochrone problem and solution. See

http://mathworld.wolfram.com/BrachistochroneProblem.html

The brachistochrone curve is the minimum transit time curve from start to finish, and is a cycloid. The cycloid initially accelerates very fast, the curve dips below the finish line, and then the acceleration lessens to zero and decelerates as the finish line is approached. See Mathematica simulation in

http://curvebank.calstatela.edu/brach/brach.htm

The story goes that someone proposed the problem to Newton, and he solved it in one day, having "invented" calculus of variations to do it.

I am hoping that someone, sometime, builds a brachistochrone roller coaster.

Bob S

 

[astrokreng]

Acceleration through a non-uniform curve can be a complex concept to understand, especially when designing a roller coaster. The first step in finding the lateral and up/down g-forces is to understand the acceleration components. As you have mentioned, the velocity of the roller coaster depends on the height, not time, due to conservation of energy. This means that the roller coaster's kinetic energy is constantly changing, but the total energy (potential + kinetic) remains constant.

To find the acceleration components, we can use the equation a = dv/dt, where a is the acceleration, v is the velocity, and t is time. However, in this case, we need to use a different equation since the velocity depends on the position, not time. We can use the chain rule to rewrite the equation as a = dv/ds * ds/dt. Since ds/dt is the velocity, we can substitute it with the equation you have already found, v = √(2g(ho-h)), where ho is the initial height and h is the current height.

This gives us a = dv/ds * √(2g(ho-h)). We can simplify this further by using the fact that dv/ds = a(s), where a(s) is the acceleration as a function of position. This gives us the final equation, a = a(s) * √(2g(ho-h)). Now, we can find the lateral and up/down g-forces by using the equation F = ma, where F is the force, m is the mass of the rider, and a is the acceleration we just found.

I understand that this concept can be overwhelming, especially with a looming deadline. I would suggest seeking help from a teacher or a tutor to better understand the concept and apply it to your project. You can also try breaking down the roller coaster into smaller sections and finding the acceleration for each section separately, then combining them to find the overall acceleration. Best of luck with your project!