Why Does a Cycloid Curve Minimize Travel Time?

[nebulinda]

I am looking for a semi-detailed description of the physics behind the brachistochrone problem. Basically, a brachistochrone is the shape of a ramp that takes the shortest time for a ball to roll down. This shape turns out to be a cycloid.

I didn't believe it when I first heard about it, and I thought a ball would take the same amount of time to roll down a cycloid ramp as a straight ramp. But after building the ramps, the ball does indeed get to the bottom of the cycloid quicker.

All the explanations that I've read are either too mathematical for me to follow, or just a sentence. I know that since the ball on the cycloid ramp starts at a steeper incline, it starts out faster, and thus reaches the bottom first, even though both balls have the same velocity at the bottom. I am still having a hard time grasping this, and I was wondering if anyone could provide a slightly more in-depth description of the physics of this situation.

 

[Bob S]

You already understand the basic physics of the brachistochrone problem; a steeper slope at the beginning gives a higher initial velocity to the ball than a straight ramp. (note: a ball has rotational moment of inertia which slows it down on all paths). It is easy to show mathematically that a parabolic curve (or some other chosen curve) would have a faster transit time than the straight-line path. The real beauty in the brachistochrone problem is that the solution developed by I. Newton found the fastest transit time of all paths. It is even more impressive that I. Newton used his newly-developed calculus of variations in 1696 to solve the problem in one day. See

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

Bob S

 

[Cleonis]

I am looking for a semi-detailed description of the physics behind the brachistochrone problem.
[...]
I know that since the ball on the cycloid ramp starts at a steeper incline, it starts out faster, and thus reaches the bottom first, even though both balls have the same velocity at the bottom.
[...]

I agree with Bob S, it does not amount to much more than the above, so you have already pretty much grasped it.

The general feature of a ramp (either with uniform slope or variable slope) is that downward motion is redirected to sideways motion.
In the brachistochrone problem it is advantagious to have most of the acceleration early in the path. You work up a lot of velocity early in the path, and then you benefit from that velocity for the rest of the duration.

Still, it's no good to cram all of the acceleration right at the start. That is, it's no good to first drop straight down, and only at the bottom redirect the acquired velocity sideways. Valuable time that could have been spent traveling sideways is then left unused.

The ramp shape with the fastest path must start redirecting vertical velocity to sideways velocity right from the start, and at the same time more acceleration must occur at the beginning of the path than later on.

 

[Cleonis]

I am looking for a semi-detailed description of the physics behind the brachistochrone problem.
[...]
All the explanations that I've read are either too mathematical for me to follow, or just a sentence.

Some remarks about the math.

Mathematically the problem is very interesting. Assuming that the problem has a unique solution, how do you get there? In the days of Jakob Bernouilly no formal apparatus was available. In those days each mathematician who solved the problem had to invent new techniques to do so.
Later a branch of differential calculus called 'calculus of variations' was developed to handle that category of problems efficiently.

In solving equations such a quadratic equations you obtain a number, or several numbers, depending on the circumstances.
Differential equations are a higher class of equations. A differential equation is for finding a function; the solution to a differential equation is a function.

In the case of the brachistochrone problem a suitable apparatus for trying to find the brachistochrone is to use the Euler-Lagrange equations. There is no guarantee that the approach with the Euler-Lagrange equations will yield an answer. Bob S included a link to a discussion on Mathworld. The mathematical ingenuity there is in the substitution that is made. There is no guarantee for that opportunity; you do not know in advance which rearrangement of the equations will be helpful in finding a solution.

In the case of the brachistochrone the trajectory can be expressed efficiently with sines and cosines. No doubt the author of the Mathworld article knew that in advance, so no doubt he was striving towards expressions that were suitable for some trigonometric substitution. (In other words, that way of obtaining the mathematical result got the job done because the author knew the solution in advance, allowing him to work towards it.)

Of course I also recommend that you look up resources about the Calculus of Variations.

 

[nebulinda]

Thanks everyone. That all has been pretty helpful.

 

[teko]

I want to know which one is the curve that gives you the fastest speed.
by this I mean to change the problem from finding the shortest time, to finding the highest speed
or do all curves from two given points give you the same speed?