Physics Puzzle: a Race Between Two Marbles
December 17th, 2007
| Categories: Mechanics, Physics Puzzles
For our weekly puzzle, here is a physics brainteaser to test your mechanics skills (did I mention already those are my favorite puzzles?)
Consider a marble which rolls from start to finish along one of two paths: either an inclined, yet straight plane (B), or a series of valleys and hills passing below that plane (A). Assuming no friction and that the marble stays on its path at all times (and doesn’t, say, hit some bump and gets thrown up in the air), which path do you predict will take the least amount of time? Or, simply put, which marble, A or B, will reach the finish line first, and why?
Added 11/2/2008: A few clarifications.
- In asking the question, I wasn’t referring to the particular paths shown in the figure, but to ANY two paths, one straight and the other curved and below the straight path.
- The marbles should be treated as point objects.
If there is no friction then both the marbles will reach the ‘finish’ point at the same time.
The marbles’ speed is limited by the gravitational force. If the path between A and B in a straight line takes x second to finish. Just make a valley so deep that the gravitational force won’t be able to speed up the other marble so it reach the bottom in x, seconds. This it might take longer time in the path which is not straight.
EDIT: “Thus it might take longer…”
Himanashu,
Nope, they won’t reach the ending point at the same time ;). Hint: use conservation of energy.
Jarle,
Are you saying marble B will definitely finish last? In that case, think again ;).
B will reach the finish line first.
The speed of A when it reaches ‘finish’ will be the same as the speed of B when it reaches ‘finish’ (according to conservation of energy).
However, since A travels a longer path, B will reach ‘finish’ first.
I think this problem has been discussed in an undergraduate introductory Physics textbook… Physics for Scientists and Engineers if I’m not mistaken.
Kelichi,
I don’t think the argument is correct. It is a long standing result of the calculus of variations that the path of shortest time is a segment of a cycloid, and not a straight line — see for instance,
http://mathworld.wolfram.com/BrachistochroneProblem.html
If your argument were correct, it would contradict this cycloid result.
I think the conservation of energy hint was meant to make us realize that, since path A is strictly lower than path B, the marble on A is always traveling faster than the marble on B. So I’d guess the answer is path A takes a shorter time than path B.
However, I’m not sure how to prove this rigorously, since time depends on the length as well, i.e. T = integral d(length) / speed, so one would have to show the increase in length does not offset the increase in speed.
If there are 50,000 or so valleys and they are 100 times deeper than the distance B travels it seems easy to see that B wins. The only dispute is by how much.
Marble B will reach faster than Marble A regardless of the gravitaion or the force in which the object is thrown.
The motion of the object in both the parts are depends on the shape and weight of the object.
So if the shape and weight is compromised between both the options. Marble B is the possible option to reach finish line first.
Path for marble (B) is inclined yet straight plane while that for marble (A)is series of valleys and hills. Assuming no friction and that the marble stays on its path at all times, marble A will reach the end point quicker. This is because a circular object takes less time to traverse over a semicircular path. This is due to ‘mating’ of the edges coming into contact. We see the same in screw threads. Thus marble B will have to travel a path of its complete circumference, while marble B has to travel a lesser distance. So, depending on the shapes of contours the effective distance can be calculated. Seeing your lousy figure, I will go for marble A as the answer. Let me know if i’m wrong.
-War$nake
I thought my lousy figures were part of the blog’s charms :D.
Let me clarify the question: I wasn’t referring to the particular paths shown in the figure, but to ANY two paths, one straight and the other curved and below the straight path.
I also meant for you to treat the marbles as point objects (or, if you wish to be politically correct about it, as spheres having infinitely small diameter).
Do you still stand by your analysis?
Nope, in that case i’ll go for marble B.
I learnt this in 4th std.:
The shortest distannce between two points is a line.
If the spheres have infinitsimal diameter, plus the path’s curves not defined, i’ll have to change my answer.
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Hmmm… This question is more complex than I originally thought. We had a demo like this in one of my physics classes and the ball corresponding to A won. This is because when the ball falls lower, it has more energy and thus moves faster.
However, after reading the comments I realize that this argument only holds if the valleys below are roughly flat. If the extra energy it has is constantly following a path that’s going practically straight up or down, it’s not getting to it’s real goal, the right side, very quickly. The time it would take would be an difficult integral that I don’t feel like figuring out right now.
With the picture you’ve shown, most of the path for A is fairly flat so I’d say that A probably would win. However, this problem can’t be solved in general without the particular path that’s traveled.
Ok, so if you have an equation for the path y=f(x)
then the time would be the integral from x0 to x1 of
(1+f’(x)^2)/(h-f(x)) dx
where h is the original height.
It’s because due to conservation of energy, mv2 = mg(h-f(x))
v = sqrt(h-f(x))
but the velocity (actually speed) is also equal to:
sqrt( (dx/dt)^2 + (dy/dt)^2)
due to the Pythagorean theorem. Set the 2 equations equal to each other and square them.
h-f(x) = (dx/dt)^2 + (dy/dt)^2
y=f(x) so dy = f’(x)dx
Substitute this in and solve for dt.
So if the slope is constant (situation of B), say -x, it’ll just be the integral of:
(1+1)/(h+x)dx
or
2ln(h+x) from the initial to final x points.
Can anyone else confirm this thread of logic?
Hi Cade,
You are correct in your method of solution, although I can’t vouch for the final answer.
This puzzle comes from the book “Mad About Physics”, but the answer given in that book is actually incorrect, and I’ve corresponded with the author on that matter. The truth is, either marble could win, depending on the shape of the valley (on f(x), as you’ve put it).
It’s one of those puzzles that seems to be intuitively obvious, but is in fact ambiguous and goes to show that sometimes physical intuition can be dangerous if left unchecked.