I 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!
 
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