Since acceleration is velocity over time, it is denoted as v'. For any particular instant in time, acceleration is force over mass at that particular time. Velocity is the antiderivative of acceleration, which is a calculus determination. The velocity is the antiderivative of it. This formula is factored for time to get the desired quantity.
Time is the only unknown, because the velocity is determined in the falling object analysis.
Fraction of time. The total time of burn represents the total energy for both the rocket mass and attached mass. A fraction of the total time was attributed to the attached mass, and average mass loss corrected for the change in mass. This method left a slight error in the calculation. The ratio of test mass to total mass is integrated with time.