Average Speed and How It Relates to Average in General

Date: 10/07/2008 at 08:43:10
From: Brian
Subject: Average speed and how it relates to average in general

The average speed of an object is distance/time.  We are taught that 
the average is found by adding up the number of items and dividing 
by how many there are.  How does this average relate to the average 
speed or are they unrelated?

Date: 10/07/2008 at 09:42:24
From: Doctor Peterson
Subject: Re: Average speed and how it relates to average in general

Hi, Brian.

Yes, these ideas are very closely related.

The essential idea of "average" in both senses is "a uniform value 
that could be put in place of all values in a collection, without 
changing the net effect".

In the case of averaging numbers (specifically what is called the 
arithmetic mean), this means that you want to find what number they 
could all be replaced by, and still have the same sum.  As an example,
if 10 people all had different amounts of money and wanted to share it
equally, they would put it all together (adding the amounts) and then
each take an equal share (dividing by the number of people).  Then
they all have the same amount, but the total is the same as it was
originally.

In the case of average speed, suppose you have been driving at 
varying speeds during different parts of a trip, but you want to know 
at what speed you could have driven to take the same amount of time, 
if you had driven at a constant speed.  To find out, you take the 
total distance driven, and divide by the total time it took.  If you 
had driven at that speed throughout the trip, you would have arrived 
at the same time.  In a sense, we have divided the distance equally 
among all minutes of the trip.

Note that in both cases you are dividing a total by a total--the 
total amount of money by the total number of people, and the total 
distance by the total time.  In fact, if you drive distinct segments 
of your trip at different constant speeds, it turns out that the 
average speed is the weighted average of the individual speeds: you 
multiply each speed by the time for which you drove that speed (which 
is called a weight) and then divide by the sum of the weights (the 
total time).

For more on some of these ideas, see:

  What Does Average Mean?
    http://mathforum.org/library/drmath/view/52809.html 

  What is the Meaning of "Average"?
    http://mathforum.org/library/drmath/view/70499.html 

  Weighted Averages
    http://mathforum.org/library/drmath/view/52724.html 

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/