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An Antiquated Math Challenge


Yoruichi-san
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One of the most famous problems in the history of mathematics, posed by Johann Bernoulli in 1696 (I won't state the official name of it, in order to make it not too google-able ;P):

Find the curve along which a particle will slide without friction in the minimum time from one given point P to another point Q, the second point being lower than the first but not directly beneath it.

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well the shortest distance between any two points would be a straight line, so I'll go with that.

unless we are trying to find a curve with the minimum friction.

Yes, but the particle is not moving at constant speed ;). It starts out with no momentum (perhaps I should have specified that, but I thought it was implied), so its momentum is dictated by gravity.

A vertical line would allow free fall :thumbsup: if it touches it initially, does that constitute sliding?

Lol...sorry it's the path it slides along, not by ;P.

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well the shortest distance between any two points would be a straight line, so I'll go with that.

unless we are trying to find a curve with the minimum friction.

Yes, but the particle is not moving at constant speed ;). It starts out with no momentum (perhaps I should have specified that, but I thought it was implied), so its momentum is dictated by gravity.

[spoiler=nice hint :blush: ]

Since the particle starts with a speed of 0, the angle at that point must be nothing, hence it is is tangent to a vertical line at the origin.

There fore the speed must reach a max value when the trajectory becomes horizontal and the angle 90 degrees.

This screams for a cycloid

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