BrainDen.com - Brain Teasers
• 0

# An Antiquated Math Challenge

## Question

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.

## Recommended Posts

• 0

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

##### Share on other sites

• 0

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.

##### Share on other sites

• 0

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.

##### Share on other sites

• 0

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

##### Share on other sites

• 0

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.

##### Share on other sites

• 0

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

##### Share on other sites

• 0

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)

##### Share on other sites

• 0

is coriolis effect to be considered?

Edited by TimeSpaceLightForce
##### Share on other sites

• 0

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

##### Share on other sites

• 0

Inertial frame of reference.

Knowing the solution is the same as solving for it .

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.