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Given a physics lab with various measurement apparatus and bouncy balls, and the ability to do mathematics, how could I experimentally determine that conservation of kinetic energy holds true for elastic collision? Let's assume I've also experimentally learned of Newton's Laws of Motion, but I know nothing else. I'd like to stay within the realms of Classical Mechanics.

Would I simply need to make balls of different masses collide and measure their speeds before and after? Or is there a more fundamental experiment I could do? For example, in the case of conservation of momentum, I could determine $F=ma$ from measuring how a weight attached on a string pulls a cart on a horizontal surface, then I think I could logically infer the third law, then use that to mathematically show conservation of momentum. Is there anything similar for conservation of kinetic energy?

I understand that I could also derive it via Noether's Theorem; however if I understand correctly, to do that I have to first know that the Principle of Least Action is true, but I haven't experimentally shown it. Is my understanding correct? And would I be able to run some experiments not having to do with bouncing balls to deduce this principle and then use Noether's Theorem? Or to deduce this principle from mechanics experiments alone I'd basically be still bouncing balls and measuring their velocities?

Regarding the idea that the "elastic" in "elastic collision" implies energy conservation -- say instead I either happen to have in my lab balls of perfect bounciness; or that I'm just aiming to show that total energy never increases upon collision, and I have in my lab balls of different bounciness.

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Regarding the idea that the "elastic" in "elastic collision" implies energy conservation -- say instead I either happen to have in my lab balls of perfect bounciness

You really cannot simply avoid this issue. An elastic collision is defined as a collision in which KE is conserved. So that fact is not subject to experimental validation, it is always true by definition.

What you can experimentally test is whether or not a given collision is in fact elastic. In other words, you have claimed to have balls of perfect bounciness. Whether or not those balls are in fact perfectly bouncy as advertised is subject to experimental verification. But whether an elastic collision conserves KE is a matter of definition, and it is true by definition.

or that I'm just aiming to show that total energy never increases upon collision, and I have in my lab balls of different bounciness.

In fact, this can be violated. It is possible to construct balls that produce hyperelastic collisions. An inelastic collision means that some energy is converted from KE to some other form of energy. A hyperelastic collision means that some energy is converted from some other form of energy to KE. Imagine, for example, a ball with a compressed spring inside which releases in a collision and pushes away.

Would I simply need to make balls of different masses collide and measure their speeds before and after?

Yes. In order to test whether or not a given collision is elastic you would need to measure their masses and then measure both their linear and rotational speed before and after the collision. There really is not another alternative.

I understand that I could also derive it via Noether's Theorem; however if I understand correctly, to do that I have to first know that the Principle of Least Action is true, but I haven't experimentally shown it. Is my understanding correct?

This cannot be derived, regardless of the principle of least action. KE can decrease, increase, or stay the same. All three possibilities are consistent with the principle of least action and Noether's theorem. But regardless of those possibilities, an elastic collision is defined as one where KE is conserved. So all you can experimentally do is determine if a collision is elastic or not by showing which possibility it represents.

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