1. May 1, 2017

### zwierz

Everybody knows what this is

Teachers very like to show it to students as an illustration of conservation laws. But this toy illustrates something less trivial also.
And the questions are
1) should students know such things?
2) are there another explanations besides the one proposed below ?

Assume that our Newton Cradle consists of three same balls each ball has mass 1. Initially the first and the second ball are at rest and those are in contact. The third ball hits the second one with velocity 2. The collision is elastic.
Let $v_i,\quad i=1,2,3$ be the velocities of the balls right after the collision. Write down the laws of energy and impulse conservation:
$$v_1^2+v_2^2+v_3^2=4,\quad v_1+v_2+v_3=2.$$
These are two equations with three unknowns. The solution $v_1=2,\quad v_3=v_2=0$ is usually demonstrated but actually this system has the continuum solutions. For example another one is
$$v_1=\frac{1-\sqrt 5}{2},\quad v_2=1,\quad v_3=\frac{1+\sqrt 5}{2}.$$
So that the model of elastic collision does not provide uniqueness of solution and thus this model is incorrect for the Newton Cradle. The explanation of this effect is as follows. The motion after the collision is very sensitive to initial position of the first and the second ball. They may not be in contact but situated very close to each other and this small distance influences heavily to the behavior of the system right after the collision. Almost the same small initial distances between the first and the second ball can give very different velocities after the collision. The result may also be sensitive to small irregularities of the balls at the contact point.

Last edited: May 1, 2017
2. May 1, 2017

### Baluncore

The balls are assumed to be in contact. The physical construction has the suspension points separated by very slightly less than the ball diameter so as to ensure contact.

Looking at the animation, I notice that the far string has a few pixels of inward slope. That actually shows the suspension points are spaced at greater than the ball diameter, which is exactly the opposite of how the cradle is made in practice. The suspension must guarantee contact. That pixel string is only visible because one vertical string is shown, but to maintain the balls in a straight line, it takes two strings in a 'V' for each ball.

Funny how the golden ratio shows up when you are not expecting it.

Last edited: May 1, 2017
3. May 2, 2017

### zwierz

the golden ratio shows up here just by accident, I could assign another initial velocity and another mass or choose another solution

Last edited: May 2, 2017
4. May 2, 2017

### Andy Resnick

Well done! Your post got me thinking about something I (initially) thought was trivial. Here's a few journal papers (I'm reading them now)

http://www.physikdidaktik.uni-karlsruhe.de/publication/ajp/Ball-chain_part2.pdf

And a good site:

As to your first question, I'll quote from one of the papers (Delaney):

"Students should see that apparently simple experiments, when closely examined, can raise a number of complicated questions. One also should be cautious about fully accepting well-established explanations of physical phenomena without carefully scrutinizing the arguments."

5. Jul 2, 2017

### mpresic2

See the play and movie, Rosenkrantz and Guildenstern are dead (Tom Stoppard) for a funny illustration of Newtons cradle on youtube