Here we discuss how position, velocity, and acceleration relate to higher derivatives.
Assuming acceleration is constant, we may write velocity and position as
where is the (constant) acceleration, is the velocity at time zero, and is the position at time zero.
These equations model the position and velocity of any object with constant acceleration. In particular these equations can be used to model the motion of a falling object, since the acceleration due to gravity is constant.
Calculus allows us to see the connection between these equations. First note that the derivative of the formula for position with respect to time, is the formula for velocity with respect to time. Moreover, the derivative of formula for velocity with respect to time, is simply , the acceleration.