[math-fun] why did everyone (except Bill) get this wrong?
Hi all, I recently posed a simple mechanics question to a bunch of university faculty. Pretty much everyone got it wrong. What I'm wondering is.. why? Am trying to explore this a bit. Might make an interesting story. Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens? a. the last ball flies out b. the last two balls fly out c. something else Turns out the answer is c. No one gets this right. OK. Bill Gosper got it right. In fact, he's completely unraveling the problem. And Neil Bickford was essentially there. Maybe I should have asked more people in math-fun. But I asked 20 university faculty (physics and engineering, mainly) and they all got it wrong. Except one guy who had to simulate it first. What's going on? - Gary
My experience with swinging two balls on the Cradle is that two balls swing out, answer B below. What's different about swinging one double-weight ball instead? The mass & angular momentum should be the same as two plain balls. The details of the collision will be different (propagation & reflection of the compression wave), but I thought the answer was determined by M & L. Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Gary Antonick Sent: Thursday, December 09, 2010 10:44 AM To: math-fun Subject: [math-fun] why did everyone (except Bill) get this wrong? Hi all, I recently posed a simple mechanics question to a bunch of university faculty. Pretty much everyone got it wrong. What I'm wondering is.. why? Am trying to explore this a bit. Might make an interesting story. Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens? a. the last ball flies out b. the last two balls fly out c. something else Turns out the answer is c. No one gets this right. OK. Bill Gosper got it right. In fact, he's completely unraveling the problem. And Neil Bickford was essentially there. Maybe I should have asked more people in math-fun. But I asked 20 university faculty (physics and engineering, mainly) and they all got it wrong. Except one guy who had to simulate it first. What's going on? - Gary _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens?
a. the last ball flies out b. the last two balls fly out c. something else
Turns out the answer is c.
You didn't say what "something else" is, so I will put in my authoritative answer: The string breaks because the ball is too heavy. Steve Gray
Steve, Sounds good to me. Anything but the tempting b. btw I thought the answer was a. Oops. When I ran the experiment it looked like just the last ball was flying out. I was totally perplexed. Last ball with a bunch of clattering. When I wrote up the problem I added the choice c at the last second. - Gary On Thu, Dec 9, 2010 at 2:09 PM, Stephen B. Gray <stevebg@roadrunner.com> wrote:
Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens?
a. the last ball flies out b. the last two balls fly out c. something else
Turns out the answer is c.
You didn't say what "something else" is, so I will put in my authoritative answer:
The string breaks because the ball is too heavy.
Steve Gray
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If you only have a single additional ball (i.e. the heavy ball and one normal ball), then I believe the normal ball should move off at (4/3)v, while the heavier ball continues moving in the same direction at (1/3)v. When you add additional balls it becomes more complex. If there were some distance between the balls it would be simpler, but because they're touching, the secondary collisions are happening almost instantaneously, making it much more sensitive to any imperfections that might affect the timing. Tom
ah. good point. in fact, this is causing a problem. I'm experimenting with actual newton's cradles (putting different sizes together) and it looks like only the last ball flies out when a heavy first ball hits. arg. the other balls move a little but *less* than they do when I drop a first ball of normal mass. very odd. On Thu, Dec 9, 2010 at 7:49 PM, Tom Karzes <karzes@sonic.net> wrote:
If you only have a single additional ball (i.e. the heavy ball and one normal ball), then I believe the normal ball should move off at (4/3)v, while the heavier ball continues moving in the same direction at (1/3)v.
When you add additional balls it becomes more complex. If there were some distance between the balls it would be simpler, but because they're touching, the secondary collisions are happening almost instantaneously, making it much more sensitive to any imperfections that might affect the timing.
Tom
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Most of my mechanics students would get this right, because I tell them to analyze Newton's Cradle, even the ordinary one, as a succession of two-body collisions. I'll pose this on the homework next time I teach the class. Thanks! Veit On Dec 9, 2010, at 12:44 PM, Gary Antonick wrote:
Hi all,
I recently posed a simple mechanics question to a bunch of university faculty. Pretty much everyone got it wrong.
What I'm wondering is.. why? Am trying to explore this a bit. Might make an interesting story.
Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens? a. the last ball flies out b. the last two balls fly out c. something else
Turns out the answer is c.
No one gets this right. OK. Bill Gosper got it right. In fact, he's completely unraveling the problem. And Neil Bickford was essentially there. Maybe I should have asked more people in math-fun.
But I asked 20 university faculty (physics and engineering, mainly) and they all got it wrong. Except one guy who had to simulate it first.
What's going on?
- Gary
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Veit, Why do you think so many faculty I asked got it wrong? You'd think mechanics is mechanics. I'm intrigued about where our intuition goes astray. And how being immersed in a field might lead one astray. Perhaps a mathematician can figure this out more easily than an engineer. Maybe that's just Bill. - Gary On Thu, Dec 9, 2010 at 10:42 AM, Veit Elser <ve10@cornell.edu> wrote:
Most of my mechanics students would get this right, because I tell them to analyze Newton's Cradle, even the ordinary one, as a succession of two-body collisions. I'll pose this on the homework next time I teach the class. Thanks!
Veit
On Dec 9, 2010, at 12:44 PM, Gary Antonick wrote:
Hi all,
I recently posed a simple mechanics question to a bunch of university faculty. Pretty much everyone got it wrong.
What I'm wondering is.. why? Am trying to explore this a bit. Might make an interesting story.
Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens? a. the last ball flies out b. the last two balls fly out c. something else
Turns out the answer is c.
No one gets this right. OK. Bill Gosper got it right. In fact, he's completely unraveling the problem. And Neil Bickford was essentially there. Maybe I should have asked more people in math-fun.
But I asked 20 university faculty (physics and engineering, mainly) and they all got it wrong. Except one guy who had to simulate it first.
What's going on?
- Gary
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Thu, Dec 9, 2010 at 12:44 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Hi all,
I recently posed a simple mechanics question to a bunch of university faculty. Pretty much everyone got it wrong.
What I'm wondering is.. why? Am trying to explore this a bit. Might make an interesting story.
Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens? a. the last ball flies out b. the last two balls fly out c. something else
Turns out the answer is c.
No one gets this right. OK. Bill Gosper got it right. In fact, he's completely unraveling the problem. And Neil Bickford was essentially there. Maybe I should have asked more people in math-fun.
But I asked 20 university faculty (physics and engineering, mainly) and they all got it wrong. Except one guy who had to simulate it first.
What's going on?
Here are two factors tat I think contribute to getting the answer b. The first is that people have done the experiment of swinging not one ball of double mass, but two balls of single mass, towards the stationary balls, and observed behavior b. Since the two balls stay together when you do this, it seems a natural assumption that welding together two balls that are staying together anyway would not change the behavior. It is surprising to me that two balls a tiny distance apart (so that the first hits the stationary balls first, and then the second swinging ball hits the first) exhibit a behavior different from the two balls welded together, yet one in which the two balls do not separate from each other. The second reason people produce the wrong answer b is that an effective shortcut in problems like this is "guess an answer, and verify that it satisfies the conservation laws". In a problem with sufficiently few degrees of freedom, there is a unique behavior that satisfies the conservation of energy and momentum, so if the guess passes these tests, it must be correct. Behavior b conserves energy and momentum, but is not the only possibility that does so, and turns out to be the wrong one. Andy
- Gary
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-- Andy.Latto@pobox.com
participants (6)
-
Andy Latto -
Gary Antonick -
Schroeppel, Richard -
Stephen B. Gray -
Tom Karzes -
Veit Elser