A common example of an application of the derivative of the position function involves the parabolic arc that objects travel when projected upward. The object's position is a function of its initial position above the ground, its initial velocity, and the negative acceleration of gravity: $s\left(t\right)=-\frac{1}{2}g{t}^{2}+{v}_{0}\left(t\right)+{h}_{0}$Where 'g' is the force of gravity (9.8 meters/sec/sec), and the values of 'v' and 'h' are the initial velocity and height of the object when it is projected.
In this example, we determine what the highest position of an object launched an at an initial velocty of 50 m/sec, from a position 10 meters above the ground.
Velocity, Acceleration, and Position.