# Acceleration through a non-uniform curve

1. Jan 23, 2010

### 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

2. Jan 23, 2010

### 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

Last edited by a moderator: Apr 24, 2017