We have seen that the four fundamental kinematic quantities are position, velocity, acceleration, and time.  We saw in the last lecture that the velocity is a measure of how fast an object’s position is changing with time.  We are now about to see that an object’s acceleration is a measure of how fast that object’s velocity is changing with time.  A little later in this lecture we will  finally get the four fundamental kinematic equations of motion.  These equations will take us far in describing and predicting the kinematics of objects.

To start us off in this lecture, let’s define the average acceleration (a vector quantity) and average x-component of acceleration (a scalar quantity — this is just the part of the object’s acceleration which points in the x-direction).

The average acceleration of an object between an initial point i and a final point f is defined to be

The part of this equation along the x-direction, called the x-component of this equation, gives the average x-component of acceleration, defined by the equation

The first equation above shows that the average acceleration is given by a change in velocity over a corresponding change in time.  Thus, if the  velocity of an object changes in any way, the object will have an acceleration.  Note that this means that if the object’s speed or direction of velocity change (or both), then there is an acceleration.  The second equation above shows us that there is an x-component of acceleration if the x-part of the motion (the x-component of velocity) changes. Note the MKS units of acceleration are m/s², as given above.

In complete analogy with velocity, we may have both an average and an instantaneous acceleration.  The average acceleration is the average value of acceleration over a given time interval.  The instantaneous acceleration is the value of the object’s acceleration (that is, change in velocity) at a  particular instant — that is, the rate that the object’s velocity is changing at that instant of time.  It will be very important to keep these definitions in mind, as well as to remember that velocity entails both a magnitude (speed) and direction.

There is a lot in this lecture, so be sure to allow yourself enough time to work on it in parts (give yourself some breathing time inbetween sections!). Also, be forwarned that Ex. 4.4 is a tough one — don’t get upset if it takes you a while to catch on to it completely! 

Example 4.1 will demonstrate a straightforward application of the definitions given above.