Sharans Equations for Velocity and Energy at Variable Fractional Heights

In the following pages, I put forth formulae which hold good for finding velocity, kinetic energy, total energy of a body which is projected vertically upwards or vertically downwards with some initial velocity u in one dimension motion, where the body has covered fraction (/) th of the total displacement h. Here the fraction (/) in my derivation is represented by for the sake of convenience. For example If a body has covered (1/2) th of total height then = 1, = 2 and = (1/2)