Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

The paths of quickest descent

I read about the brachistochone problem, the cycloid and the paths of quickest descent, but I'm not sure I understand what the premise is.

I read in one place that an object is assumed to be sliding down a frictionless surface. If it is indeed frictionless, wouldn't it slide down a slope of 45 degrees as fast as it will fall straight down? So how would first allowing a more vertical drop enable it to reach the base quicker?

Is mass or momentum considered in this problem?

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luca-deltodesco

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Registered: 2006-05-05

Posts: 1,470

Re: The paths of quickest descent

if its frictionless, all it means is at the bottom of the slope it will have the same speed as if it had falled straight down, but it will reach the speed over a longer period of time

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mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: The paths of quickest descent

you mean 2 dimensional speed? or just vertical speed?

If it reaches that speed over a longer period of time, uh...wouldn't that not be fastest?

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A logarithm is just a misspelled algorithm.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: The paths of quickest descent

If the object drops vertically down, its acceleration is g. If it slides down a surface, the surface will exert a normal reaction on it; the vertical component of this normal reaction will act in opposition to gravity – and hence the vertical acceleration of the object must now be less than g.

The shortest time for the object to reach the baseline is of course to drop vertically down to the point directly below. However, if the object has to reach a point on the baseline that is not directly below, it will have to slide along a surface – and this will always be slower than dropping directly down. The best thing in this case is to choose the surface that will minimize the time taken to slide down – and it turns out that the inverted cycloid is such a surface.

Last edited by JaneFairfax (2007-05-26 19:45:56)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: The paths of quickest descent

you mean 2 dimensional speed? or just vertical speed?

It’s vertical speed. What Luca means is that the final vertical speed in both cases (dropping directly down and sliding down a surface) is the same (namely √(2gh where h is the vertical height), but because the object accelerates more slowly in the sliding-down case, it attains its final vertical speed in a longer time than in the directly-dropping-down case.

Let the vertical acceleration in the sliding-down case be a.

Time to reach baseline by dropping directly down is

Time to reach baseline by sliding down surface is

If it reaches that speed over a longer period of time, uh...wouldn't that not be fastest?

Sliding down a surface will never be faster than dropping directly down. However, if the object has to reach a point on the baseline that is not directly below, it will have no choice but to slide down a surface. In the latter case, the brachistochrone curve (cycloid) enables it to make the trip in the quickest time – but even this will not be quicker than dropping directly down.

Last edited by JaneFairfax (2007-05-26 20:09:36)