The basic rocket equations were provided by the Jet Propulsion Laboratory. The velocity equation was derived independently. Rocket Applied to Falling Objects for Proof. The rocket is used to determine the amount of energy in the falling objects by attaching the masses and accelerating up to the required velocities. The amount of burn time for the rocket is determined.
Time represents fuel use, since fuel is used at a constant rate.
However, these fractions need to be corrected slightly, because the rocket mass changes as fuel is used. The exhaust mass equals burn time times rate of mass loss. Actually, the average should be a little less than half, but dividing by two is a very close approximation, which is adequate for the purpose. An exact procedure is shown on the explanation page. Averaging is done here to create the clearest possible logic, so someone doesn't assume I'm pulling the wool over their eyes.