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An Antiquated Math Challenge

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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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Posted · Report post

What is the value of g ? Is the marble rolling?

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Posted · Report post

You can use g to represent the gravitational constant, the answer should be in terms of g. I'm not sure what you mean by the other question, it's particle sliding down the path you choose without friction.

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Posted · Report post

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.

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Posted · Report post

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

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Posted · Report post

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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Posted · Report post

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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Posted · Report post

okay i think i got ya y-chan.

the equation for gravity is d = 1/2*g*t^2

so i would think the equation for the curve would be...

y = 2/(g*x^2)

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Posted (edited) · Report post

is coriolis effect to be considered?

Edited by TimeSpaceLightForce
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Posted · Report post

This curve is called the Brachistochrone curve, which is an inverted cycloid

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Posted · Report post

Inertial frame of reference.

Knowing the solution is the same as solving for it ;).

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