This problem is more interesting than I thought. First of all, the whole thing doesn't actually work. at least with my cheapo cradles. I swung a mass 2m into 4 mass m balls and it appears as if only the last ball moves. The 4th ball moves but not more than it normally would will all balls of same mass. Second, the constraint issue. There's two equations and five balls. If we didn't know anything else the solution would be indeterminate. So we must know something else. Which we do: the balls are in a line. (They're not arranged like bowling pins or in a semicircle to connect with the first ball all at once, for example.) Therefore (I still don't iunderstand this) the collisions *must* be able to be considered in pairs. Third- the gap between balls. I still don't understand this at an atomic level. What if these were cubes? In a vacuum? etc. On Fri, Dec 10, 2010 at 9:13 AM, Cordwell, William R <wrcordw@sandia.gov> wrote:
[posted per request]
I thought of it as all of the balls having very slight gaps between them. The initial, heavy, ball hits the first stationary ball in the array, knocking it forward with 4/3 of the initial ball's speed, and the initial ball continues forward more slowly (at 1/3 its initial speed). The first ball in the array acts like an initial ball on the (n-1) remaining members of the array, performing rapid (with almost unnoticeable delay) consecutive (almost) elastic collisions, propelling the final ball in the array out with the 4/3 v0. The initial, heavy, ball, catching up to the now stationary first ball, does the same thing, but starting with the reduced 1/3 v0. It should give a nice, simple geometric series for the total kinetic energy (and for the momentum, as well).
Since, when all of the balls are the same mass, and if two balls are launched towards the remaining array, then two balls come off, there must be, effectively, a small gap between the initial pair. It would be interesting to know if the gap is actual, or if the shape (so that the balls touch at almost just a single point) allows compression/behavior equivalent to having a gap.
Nice problem!
Thanks, Bill
-----Original Message----- From: gantonick@gmail.com [mailto:gantonick@gmail.com] On Behalf Of Gary Antonick Sent: Thursday, December 09, 2010 7:53 PM To: Cordwell, William R Subject: Re: [math-fun] why did everyone (except Bill) get this wrong?
Hi Bill,
Your guess is right on. You're not in the group of 20.
Since you got it right.. how did you think about the problem? Can you see why the other guys went astray?
- Gary
On Thu, Dec 9, 2010 at 2:26 PM, Cordwell, William R <wrcordw@sandia.gov> wrote:
Dear Gary,
So, if I were somewhat guessing, I'd say that the end ball comes off with speed 4/3 v0, followed almost immediately by the 2nd-to-the-end ball at maybe 4/9 v0, etc.
Am I in with the group of 20?
Thanks, Bill C.
-----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
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