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I'd like to conduct some measurements with my pupils in the high-school I'm teaching in, but I ran into some conceptual problems. I'd like to measure the (approximately constant) speed $v$ of an electric train running on a circular track by measuring the time taken by the train to cover different distances.

The setup is this:

  • each distance and time measurement is taken a single time, i.e. it's a single direct measure;
  • for each couple $(d_i,t_i)$ of measures I can calculate the speed $v_i=d_i/t_i$, i.e. I obtain an indirect measure;

My question is this: How can I correctly (following error theory) evaluate the "true" speed $v$? Can I take the mean of the $v_i$'s? What about errors? I suppose that single measures have an error related to the measuring tool, i.e. if the stopwatch shows the centiseconds, the times shoud have an error of 0.01 s and if the ruler has millimeter precision, the distances shoud have an error of 1 mm.

I'm sorry but I don't have a sound background in error theory so I don't know in theory how to handle errors in indirect measures obtained by single measures.

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  • $\begingroup$ If the stopwatch shows centiseconds, the largest error contribution factor will be the reaction time of the person pressing the button, not the watch itself. For any visual scale (like a ruler), a good rule of thumb is that the error incurred by reading from it is half the scale. For combining errors, see e.g. this post. For the "mean" - do you know what a standard deviation is? $\endgroup$ – ACuriousMind Jun 16 '15 at 10:36
  • $\begingroup$ If I set this expt as a practical I would expect my students to graph distance as a function of time and calculate the velocity by drawing a straight line to fit the points. Errors can be roughly estimated by the range of straight lines that give plausible fits. You could then use a linear regression app to calculate the best fit gradient and compare it to the students' results. Some linear regression apps will also give standard errors for the gradient and intercept. $\endgroup$ – John Rennie Jun 16 '15 at 11:05
  • $\begingroup$ @JohnRennie, The problem is that the students don't know what a linear regression is; then again the problem is about error theory, not regarding the actual values: i'd like to give insight to the problem rather than using prefabricated applets. $\endgroup$ – marco trevi Jun 16 '15 at 11:26
  • $\begingroup$ @ACuriousMind, the post you linked me has another kind of setting, i.e. multiple measurements. I know about standard deviation, but how would you use it in this context? $\endgroup$ – marco trevi Jun 16 '15 at 11:28
  • $\begingroup$ I strongly recommend you first read and work through an introductory text on statistics and error analysis. NOt only will that give you a better feel for what sources of error are in play, and how to account for them, but if you don't learn stats, your students are going to ask questions that you can't answer. $\endgroup$ – Carl Witthoft Jun 16 '15 at 15:21

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