Is there a way to build upon our average kinematic quantities between two points in time to describe the motion of an object an any instant in time?  Yes, by making the time interval for our average calculation smaller and smaller.  Initially, we might calculate the average velocity for an object over a ten second interval.  We could redo the calculation for a one second interval. But what happens as the time interval gets increasingly smaller?  As 1.0 s becomes 0.1 s, then 0.01 s, the approximation of the velocity provided by the average velocity value gets closer and closer to the actual value of the object’s velocity at a single instant.  What we are left with is the instantaneous velocity of the object.  If we start with an object’s average acceleration between two points in time, we can use the same procedure to make the time interval that we use in our calculation smaller and smaller.  As the time interval decreases toward zero, we are left with the object’s instantaneous acceleration at a single point in time.

We can use the relationship between position, velocity, and acceleration to derive a set of equations which describe an object’s motion as a function of time.  These equations of motion allow us to calculate where an object and how it is moving at any point in time during the interval where the equations describe the object’s motion.  To start with, we will look at the simplest types of motion, uniform motion where the velocity of the object doesn’t change as it moves and motion with constant acceleration.  As we will see, the equations of motion for these two relatively simple way that objects can move can accurately describe a wide variety of objects as they move in the real world.