If we let y = x(t) be the position of a particle with respect to time, then the from t =a to t = b can be found using the formula

##ave. vel. =(x(b)-x(a))/(b-a)##

Velocity is simply the change of position with respect to time. The numerator, ##x(b)-x(a)##, represents the change in position, and the denominator, ##b-a##, represents the change in time.

Let’s look at a position-time graph and calculate the average velocity over a time period.

Given the position-time function ##x(t)=-x^2+16## and the time interval from t=1 to t=3 we calculate the average velocity as follows

##ave. vel. =(x(3)-x(1))/(3-1)=(7-15)/2=-8/2=-4##

Notice that this is the slope of the secant line to the graph of x(t) through the points ##(1,15) and (3,7)##.