Mapertius's Principle and 2nd Newton's law

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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?

Handwritten derivation on lined paper: a sketch of a parabola-shaped region with axes labelled x, y, V0, L and -V0, then the equation V = (V0^2 - 2ax)^(1/2), followed by a double integral from 0 to L and worked algebra ending in (2 (V0)^3)/(3a) labelled Area between curves.
Handwritten derivation on lined notebook paper using kinetic energy, average displacement and potential energy relations, with a final expression ending in a 1/3 coefficient and a note to continue on the next page.
Handwritten derivation in blue ink on lined paper: 1/3 mV^2 = |F|W/(4 V_0^2), then 4/3 m V_0 V_0^2 = |F|W, then 4/3 m V_0^3 over |W| = |F|, with W = 2/3 V_0^2/a, leading to 2 m·a = |F|, with the question 'Is that 2 should be here?' written beside the 2.


I appreciate any help you can provide!
 

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