Which Method Best Determines Overall Average Velocity?

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Discussion Overview

The discussion revolves around the appropriate method for determining the overall average velocity of an object based on multiple timed trials over a set distance. Participants explore different approaches to averaging velocities and the implications of each method on the final result.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether averaging the average velocities from multiple trials is appropriate, suggesting that averaging the times first may yield a more accurate overall average velocity.
  • Another participant proposes that if the variation in measurements arises from time, then averaging the measured times and calculating the average velocity from that is advisable, but notes that the approach may depend on the experimental setup.
  • A different viewpoint emphasizes that if time intervals are equal, averaging the means is acceptable; however, if they are not, total distance over total time should be calculated to avoid misrepresentation of the results.
  • A participant argues that the two methods of averaging yield different results depending on the context of the measurements, providing examples to illustrate when each method would be appropriate.

Areas of Agreement / Disagreement

Participants express differing opinions on the best method to calculate overall average velocity, indicating that there is no consensus on the issue. Multiple competing views remain regarding the implications of averaging methods based on the nature of the measurements.

Contextual Notes

Participants highlight the importance of understanding the setup and conditions under which measurements are taken, as well as the potential for different averaging methods to yield different results depending on the context.

Gersty
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We were told to time the linear motion of 3 objects over a certain distance three times. Then we were to use those times to calculate the object's Vavg for each trial. Finally we were supposed to average these averages to get the overall Vavg for each object.
My question is: Is this an appropriate way to get the overall average velocity of the object over three trials. Wouldn't it be better to average the times and do distance over time with the average times?
 
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If your variation comes from the time measurement (and the objects have the same average speed in all repetitions), average the measured times (as those have well-defined uncertainties) and calculate an average velocity based on that.
If your variation comes from something else, it depends on the setup.
 
If the time intervals are equal in each of the three cases then you can take a mean of the means. If not, then you need to calculate total distance over total time or the three measurements don't get equal weighting.
Imagine that there is a wide variation in instantaneous velocity and that you do one measurement over a long interval and two others over very short intervals. The single average over a long interval will have reduced the error considerably but the two extra measurements could be well adrift and they could distort the overall answer out of all proportion with their significance if you take a mean of the means..
 
Gersty
The two methods give different results - one is correct, and one is not, depending upon the setup.

If you are measuring the time to travel a set distance, then the trial times are averaged before calculating the average velocity from the distance / time(avg). The average of the velocities of the trials gives the incorrect result.

If you are measuring the distance traveled in a set time, then the trial distances are averaged before calculating the average velocity from distance(avg) / time. Or in this case you could calculate each trial velocity and average all the trials.

The problem is often asked as follows.
A car travels 60 mph for 1 hour and 30 mph for one hour, what is the average velocity?
Ans, 45 mph. [ 60 miles + 30 miles ) / 2 hours ].
. Or [ ( 60 mph + 30 mph ) / 2 = 45 mph ] with velocity averages.

Which is not the same as; ( this is your experiment )
A car travels 60 miles in one hour and 60 miles in 2 hours, what is the average velocity?
Ans: 40 mph - total distance / total time
. ie [ (60 miles + 60 miles ) / 3 hours ].
Or : 40 mph - by averaging velocities for equal set time intervals
. ie .60, 30, and 30 mph each for 1 hour Avg = ( 60+30+30 ) / 3 = 40 mph
Or : 40 mph - set distance / average time
. ie avg time = 1 hour + 2 hour = 1.5 hour
. Set distance ( 60 miles ) / avg time ( 1.5 hours ) = 40 mph
 
Last edited:

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