An Antiquated Math Challenge

[Yoruichi-san]

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.

[Brainy Binary]

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

[Yoruichi-san]

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.

[phil1882]

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.

[BMAD]

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

[Yoruichi-san]

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 if it touches it initially, does that constitute sliding?

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

[BMAD]

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 ]

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

[phil1882]

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)

[TimeSpaceLightForce]

is coriolis effect to be considered?

Edited July 7, 2013 by TimeSpaceLightForce

[k-man]

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

[Yoruichi-san]

Inertial frame of reference.

Knowing the solution is the same as solving for it .