Average Speed and How It Relates to Average in GeneralDate: 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/ |
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