Just to give this thread some closure, http://www.physikdidaktik.uni-karlsruhe.de/publication/ajp/Ball-chain_part1.pdf is a paper which explains (in the special case of Newton's cradle) why a many-at-once collision theoretically has infinitely many solutions, but why in real life we only see one...
Dear A.T.,
Thank you for the detailed response. How do your four principles explain Newton's cradle? The Wikipedia article consistently reiterates that a 3-at-once collision has many possible solutions (something which I did not know before) and that all the animations of Newton's cradle...
I still don't get it. Does B2 (the left ball) transmit all of its momentum to B1 (the middle ball)? Does this momentum "pass through" into B3 (the right ball)? How much momentum does B2 (left) receive from B1 (middle)? Does B2 (left) receive, indirectly, any momentum from B3 (right)?
For the...
UltrafastPED: I'm not sure I understand. Could you explain your answer in the context of the following setup?
Let B1 be a ball of radius r1 = 1 and position x1(t) = (t, 0).
Let B2 be a ball of radius r2 = √2 − 1 and position x2(t) = (t − 1, 2t − 3).
Let B3 be a ball of radius r3 = 2 and...
Hello everyone,
I'm trying to find the exact velocities of three balls in the plane after they collide elastically. I'm assuming arbitrary positive masses and arbitrary positive radii. Of course, three balls can collide in many ways:
in an equilateral triangle: one coming from north, one...
I thought of this today while eating apples.
Suppose we have two arbitrary sets X and Y and a surjection g:X→Y. We seek an injection f:Y→X. Each element of Y has at least one pre-image in X, but there might be more than one; the nonempty sets X_y = \{ x\in X : g(x) = y \} are subsets of X...
It's quite a cute proof by induction, actually, if you can prove (or already know) that \sum_{k=1}^n F_k^2 = F_n F_{n+1}.
Let's assume F_1 = F_2 = 1 and F_{n+2} = F_{n+1} + F_n for natural numbers n. Since the recursive relation refers to two lesser naturals, we should proceed with strong...
It appears to be defined thus. See Beardon's Iteration of Rational Functions, pp. 40-41.
Also, http://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappings.
Thanks for your time!
Oh, I think I've gotten somewhere.
Suppose P(a) = a. Let φ be a Mobius map. With b = φ(a), the derivative of φPφ-1 at b is φ'(P(φ-1(b))) P'(φ-1(b)) / φ'(φ-1(b)) = φ'(P(a)) P'(a) / φ'(a) = φ'(a) P'(a) / φ'(a) = P'(a).
So we can get P' at a fixed point a by evaluating the derivative of φPφ-1 at...
I am reading a recent (2003) paper, "Fatou and Julia Sets of Quadratic Polynomials" by Jerelyn T. Watanabe. A superattracting fixed point is a fixed point where the derivative is zero. The polynomial P(z) = z2 has fixed points P(0) = 0 and P(∞) = ∞ (note we are working in \hat{\mathbb{C}} =...
If C is a simple closed contour such that w lies interior to C, and n > 1, then
\int_{C} \frac{dz}{(z-w)^n} = 0. I'm confused because the function f(z) = (z-w)^{-n} has a pole at w, so it isn't holomorphic, but the integral is still zero. The Cauchy-Goursat Theorem says that if f is...