To derive a kinematics formula for a ball thrown upward, one can utilize the standard equations of motion, specifically the SUVAT equations. The initial velocity (u) is known, and the acceleration due to gravity (g) acts downward. At the peak of the throw, the final velocity (v) is zero. By applying the equation v^2 = u^2 + 2as, where 's' is the displacement, one can calculate the maximum height reached. Understanding these principles allows for accurate predictions of the ball's motion.