Undergrad Mapertius's Principle and 2nd Newton's law

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The discussion focuses on deriving the second Newton's law using the Mapertius principle in the context of perfectly elastic collisions and free fall acceleration. The original poster is encountering a persistent issue with obtaining a coefficient of 2 in their final calculations. Participants are asked to provide insights or explanations for this discrepancy. The conversation aims to clarify the application of principles and calculations involved in the derivation process. Assistance is sought to resolve the confusion surrounding the coefficient.
MP97
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Hi, I have been working on deriving the second Newton's law from the Mapertius principle applied to a perfectly elastic collision from a free fall acceleration problem. These are my calculations, but I keep getting a coefficient of 2 in my final answer for some reason. Could someone explain to me what I am doing wrong?

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I appreciate any help you can provide!
 
Topic about reference frames, center of rotation, postion of origin etc Comoving ref. frame is frame that is attached to moving object, does that mean, in that frame translation and rotation of object is zero, because origin and axes(x,y,z) are fixed to object? Is it same if you place origin of frame at object center of mass or at object tail? What type of comoving frame exist? What is lab frame? If we talk about center of rotation do we always need to specified from what frame we observe?

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