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http://img-9gag-fun.9cache.com/photo/aX9zqD2_460s.jpg [Broken]

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- Thread starter micromass
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http://img-9gag-fun.9cache.com/photo/aX9zqD2_460s.jpg [Broken]

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- #2

jedishrfu

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B

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NOT 2 BB

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Without specific data and judging solely from the diagram I intuitively tend to vote for B.

- #5

Borg

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If the start and stop height differences are the same, wouldn't the answer be C?

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- #7

jedishrfu

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NOT 2 B

What was the question again?

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If the lines are frictionless then it shouldn't matter

- #9

Borg

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HINT:

Well, when you put it that way...

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If the lines are frictionless then it shouldn't matter

Suppose the blue line had a smaller slope at the beginning and end. The ball could take a long time to get across.

And, if the red line had a really large dip, it could take a long time to get across.

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- #12

jedishrfu

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https://lecdem.physics.umd.edu/question-of-the-week-archive/101-qotw-001-with-answer.html

- #13

Borek

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It's easy enough to compare a straight line down a slope with the limiting case of freefall, followed by horizontal motion. If the height of the slope is ##h## and the horizontal distance is ##d##, then the times are:

##t_1^2 = \frac{2(h^2 + d^2)}{gh}## (for the straight line)

##t_2^2 = \frac{2(h + d/2)^2}{gh}## (for free fall plus horizontal)

The straight line takes longer when:

##h < \frac{3d}{4}##

But,of course, the brachisrochrone is always faster than both.

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1) Suppose we have two paths with one ball on each path. The both start at height ho at xo and finish at height h1 at x1. h1 is less than ho. xo, and ho are taken to be the 0 datum. Positive h is downwards.

2) Suppose further that we know height profile of the paths h(x) between xo and x1.

How do we calculate which ball will reach x1 first?

My solution:

Neglecting friction and all other interesting physics, we have a relationship between the height and the velocity of the ball, v(x) = √[2*g*h(x)]

We can estimate which ball will get to the finish line first by computing its average velocity in the x direction. Since we know the paths of the ball, we can average over [xo,x1].

Taking the angle θ to be measured between positive x and tangent to h(x), the average can be computed as: {∫√[2*g*h(x)]*cos(θ(x))dx}/(x1-xo) from xo to x1.

In this case we would have to show that the product of those expressions is greater than a case where h(x) is constant (as in case 1 for example).

Now, that integral might be a pain because we have a general function h(x) and a trigonometric function. I thought to myself "How can we relate h(x) to θ?" Ah ha! We know that h(x) = ∫{dh(x)/dx}*dx, and we know that the slope of h(x) can be expressed as the tangent of θ at a point x. So we have an equation in terms of θ, only:

average(v(x)_x) = { ∫√[2*g* (∫tan(θ(x))*dx)] * cos(θ(x))dx }/(x1-xo)

And I hate to quit but this is as far as I got before looking up this integral in mathematica and seeing how many terms are in the solution. The only way I know how to do this by hand is through integration by parts, which would have to be done twice since the angle is parameterized by x.

Anyway, you can see that if h(x) is constant, then the cosine term goes to 1, the tangent term becomes h(x). You need to show that h(x)*cos(θ) on average is greater for the slopey case than h(x) in the flat case.

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Everyone else seems to be seeing a diagram. Not I.

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θOk the answer to this problem was bothering me, so I did some math:

A simpler problem, but relevant to this question would be:

Take two balls with the same initial velocity, to travel a distance ##d## along the track. The first ball goes horizontally. The second ball goes down and up an incline (assume two symmetric straight lines). What's the relationship between the times, the initial velocity and the angle of the incline? What is the optimum angle?

- #18

Borek

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The straight line takes longer when:

##h < \frac{3d}{4}##

Actually now that I think about it, it is trivial - let's say we go down, horizontal, up (sides of a rectangle, or more precisely, sides of a right trapezoid). There always exist a rectangle in which going down takes exactly as long as the horizontal leg in the "flat" case. That means going through the sides of the rectangle will take a bit longer than two times the flat case. That in turn means not every path below the flat one is guaranteed to be faster - or, in other words, B is not guaranteed to be faster than A, and to be sure which is faster we need to know exact shape of B and calculate the time it takes to travel both paths.

Or am I misunderstanding something?

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Actually now that I think about it, it is trivial - let's say we go down, horizontal, up (sides of a rectangle, or more precisely, sides of a right trapezoid). There always exist a rectangle in which going down takes exactly as long as the horizontal leg in the "flat" case. That means going through the sides of the rectangle will take a bit longer than two times the flat case. That in turn means not every path below the flat one is guaranteed to be faster - or, in other words, B is not guaranteed to be faster than A, and to be sure which is faster we need to know exact shape of B and calculate the time it takes to travel both paths.

Or am I misunderstanding something?

Yes, you have to calculate in each case.

The problem here involves an initial velocity, which complicates everything. If the initial velocity is low, then many paths below the horizontal will be faster. The optimum path will be related to the brachistrochrone, but the initial velocity messes things up. But, steeps paths, curves, straight inclines (down and up) will all be better than going along the flat.

As the initial velocity increases, it should still be possible to beat the flat path, but only by going down a short distance. I haven't tried the maths (yet) but the equation involves the relative (proportional) increase in average velocity against the proportional increase in distance. The faster the initial velocity, the smaller the proportional increase you get by going down. If the ball is going fast enough, then nothing will be significantly faster than just going along the flat.

The answer is some paths below the horizontal beat the horizontal path and some do not. The key factors are the initial velocity and the force of gravity (as well as the shape of the path, of course).

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http://img-9gag-fun.9cache.com/photo/aX9zqD2_460s.jpg [Broken]

Personally, I would have to find out the time it takes for the blue ball reach N, O, then P and similarly for the red one to reach N, X, O then P before being able to make a comparison.

HINT:

This is unrelated to the problem though.

Is this one of the unsolvable physics problems you once said you did ask on yahoo groups and you sometimes got (incorrect) answers ?

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- #22

Borek

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From these numbers it should be easy to see that in general a journey with variable heights will take a longer time to complete.

Can take longer, not "will take longer". Actually the fastest trajectory (mentioned earlier, called brachistochrone), is curved with a lowest point somewhere in between.

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Without further information they won't move at all. (Newton 1)

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Borek

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I feel like whether the curve counts as "deep" or "shallow" depends not only on its shape, but also on the acceleration involved.

In my example (the one with trapezoid) time required for a vertical leg to finish is [itex]t = \sqrt{\frac{2L} a}[/itex] - so even for a shallow trapezoid there exist an a, L combination that gives a required time of travel.

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I'd like to travel a long the curved ones.

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Borek

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Then how come brachistochrone exists, if time on all curves is identical?

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