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Assuming there is a series of delta time and displacement measurements, there are two way to calculate the average velocity I can think of:

  1. Divide total displacement by total time.

  2. Average of all momentarily velocities.

I see a lot of school text books using method 1, but can't understand why is it the correct one.

Artium
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2 Answers2

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If you use method 2, and the delta time measurements are not all the same, you will need to use a weighted average. The weight is equal to delta time for each measurement, so it will end up equal to total displacement over total time.

Arthur Fabian
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  • I do not understand, why will I need to use weighted average? I have a series if momentarily velocities and want to calculate the average of them. – Artium Jun 02 '18 at 06:41
  • @Artium An average is not simply the sum of a list of quantities divided by the number of quantities. Some quantities are more important or consequential than others. The $\delta t$ of each small measurement tells the importance of each velocity measurement. – Bill N Jun 02 '18 at 14:25
  • Delta time is the weight because each velocity is in effect for a different length of time, namely delta time. If you drive 60 mph for 2 hours and 50 mph for 1 hour, the average will be (2 h * 60 mph + 1 h * 50 mph)/(3 h) = 56.666... mph, this opposed to simply the 55 mph that you would get by ignoring the length of time spent at each velocity. – Arthur Fabian Jun 02 '18 at 16:33
  • @Artium to truly calculate the average velocity, you need to take the average of all velocities at every instant of time. Thus, the longer intervals of momentary velocity have a higher weight (since they have more "instants" of time). To illustrate, suppose you travel 1 m/s for 1s and then 10 m/s for 1,000s. Intuitively, your average velocity must be closer to 10 m/s than 1 m/s – Burrito Jun 02 '18 at 20:32
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If you treat all your small delta trips as one trip and want to determine its average or typical speed, you should use method 1.

If you treat each delta as a separate trip and want to determine an average or typical speed of a trip, you should use method 2.

If all your deltas are the same, the results will be the same as well. Otherwise, the results would be different.

For instance, if one of your deltas is much longer than other deltas and the speed in that interval happens to be zero, the average speed of your whole trip calculated by method 1 would be low, fairly reflecting the situation. If, on the other hand, you use method 2, the contribution of the long zero speed delta won't be properly weighted and the average speed of the whole trip would be exaggerated.

V.F.
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