SCM 541: Describing the Physical World

Lesson 7: Instantaneous Velocity and Acceleration

In Lesson 6, the concept of average velocity was explained and its calculation was based only on the distance between the beginning and end point and the time it took to cover that distance.  The result gives you no information about the actual path you took or what changes occurred in your speed along that path.  In the example where you ended up back at your starting point and both the displacement and average velocity was 0 (in ft and ft/s, respectively), you have absolutely no information about how far you actually walked or has fast you did the walk.

Yet, when you broke down your excursion into two components (going and coming), there was a displacement (+10 ft and - 10 ft) and an average velocity (+2 ft/s and -1 ft/s) associated with each component.  If you further broke down one leg of the trip...i.e., running 8 ft in 3 seconds, crawling 2 ft in 1 second, with a stop between for 1 second...you would have three additional sets of displacement and velocity vectors that more completely describe the trip.  

What would happen if you broke down each of these new segments further to very short time intervals? You would improve the completeness of the description and be able to graph the magnitudes of the displacement and velocity vectors as a function of time for each short time interval along the path.  In the limit of infinitesimally small  time intervals, each velocity vector becomes the instantaneous velocity.

Since the instantaneous velocity can change with time, you will need a quantity that describes that change.  This quantity, the time rate of change of velocity with time, is called the acceleration, and it, too, is a vector. Like the time rate of change of displacement (i.e. velocity), the mathematical definition is  the difference between the initial velocity vector and the final velocity vector divided by the time interval during which the change occurred.:

a = (v2 - v1) / (t2-t1)

Like the case of the definition of velocity, this is an average acceleration for the time period t2-t1.  The acceleration could have changed up and down during time period, but you have no information about that.  Analogous to the relationship between average and instantaneous velocity, you could sample the acceleration along the path in smaller time intervals until you reach the infinitesimally small limit. In this limit, each  acceleration vector becomes the instantaneous acceleration.

What do we call the time rate of change of acceleration?  Let's not go there (Whew!).  The applications of general physics are usually limited to constant acceleration, in which the average acceleration equals the instantaneous acceleration. So you only have to concern yourself with a constant, average acceleration.

The elements of this lesson are contained in the following links.  Proceed to each element below in the order listed. You may want to bookmark this page so you can find these elements quickly in the future.

                      Principles by Discovery

                       Principles by Analysis

                       Discovery Revisited

                       A Problem for Discussion

                 Homework

When you're finished the elements of this lesson, go the the "Assignments" section of the course for the assignments for Lesson 2.

© 2002 Barry L Lutz