Netwon’s Cradle is a fun desktop toy. You can create all kinds of cool patterns with the motion of the balls, but let’s focus on the simplest. When a ball is raised and released, it hits the 4 stationary balls and eventually transfer all of its momentum to the final ball.
Here’s my question, especially to you high school physics teachers. Why does this happen?? Why does only one ball get knocked up? Why not two, or more?
The intelligent move would be for me to look this up and make sure I’m right, but where’s the fun in that, so this is off the cuff.
The cradle needs to conserve momentum (no unbalanced external forces) and energy (all collisions are basically elastic).
If it were just momentum, then it would be fine for two balls to fly out at half the speed. However, that configuration would not conserve energy! (KE = 1/2 mv^2) Having both of these constraints really limits the patterns the balls can create. I expect that the motion you observe in each case is the only way that the collisions can conserve momentum and energy.
Let me know if that makes sense, or if I need to explain more about collisions or about either conservation law
I’m asking you in class. You don’t have time to look it up!
I think there are 2 problems with that explanation:
You’re right. If one ball hits with velocity v_0, even if you allow two balls come off the end with different velocities v_1 and v_2, the two equations of conservation of energy and momentum give exactly one solution, v_1 = v_0, v_2 = 0.
But what if you allow any number of balls to move with any velocity? The system of 2 equations is underdetermined. For example, the original ball could move back with velocity \frac{3v_0}{5} and all other balls move forward with velocity \frac{2v_0}{5}. (Wolfram alpha).
So why doesn’t this happen?? You can look it up but I’ve not found a very good explanation online. Wikipedia mentions the issue above but it doesn’t cover the alternative explanations as well.
Never underestimate the power of allowed states! They not only allow us to say “penetration into the classically forbidden region,” but are also a very powerful tool for understanding what will happen. I could show you the equation for why momentum is conserved, derived from Newton’s 3rd Law. That could then be applied on a force-by-force basis to explain what I said earlier, but still only leaves an expression for allowed states, does not pick one.
My remaining guess is basically Occam’s razor. Either the other states are only possible at higher energy levels (as is common in quantum physics), or it could be that the solutions we see are somehow the simplest solutions.
After looking at your Wolfram work: Is that the only other real solution? I think it might be. And then we would just be looking for a reason for that one not to happen.
Predictive power but unfortunately maybe not explanatory.
I’m not sure. I think solving general systems of nonlinear equations numerically is hard and I’m not sure they do fully do it.
I think an explanation* must be in some ways reductionist. I think here we have to consider the way the force actually propagates through the balls as compression waves at the speed of sound. They act like springs.
*There is a deeper issue here of what is an explanation and what is simply a property. Occam’s razor is a property, obviously. What about Newton’s laws? How can we assign causality to physics? All particles are excitations in QFT. So what does the rest mean?
Perhaps we want to find a minimal set of axioms and call these causes. What defines minimal? And does minimal imply fundamental?
Sorry to harsh your “the explanation must be fundamental” vibe, but I have some new information.
My students have been working on inelastic collisions and I was reminded of something. An elastic collision between two objects is a system of equations:
mv + mv = mv + mv and
.5mv^2 + .5mv^2 = .5mv^2 + .5mv^2
Where there are 4 different(maybe) v’s and 2 different(maybe) m’s, and we know the starting values for all.
This system is a circle and a line which has at most two solutions. They are typically the starting velocities and the ending velocities. For two objects of equal mass, the objects will always trade velocities, it is the only possible solution.
From there, we can consider Newton’s cradle to be a system of collisions that happen one at a time. This is true because maybe there is a small space between the balls and definitely because they are not perfectly rigid, so it takes time for them to begin moving once struck. In any case, you end up with a series of moving balls striking still balls and becoming still. If you trace this motion until it reaches the other side, you arrive at all the results that we actually see.
I think it is possible that in a perfect system, some of the weird solutions that we found could happen? And that the reason they do not, is because our imperfect system does not allow all 5 balls to be colliding together at the exact same time.