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In some books of Physics in Italian language, they write that the instantaneous velocity $v$, is:

$$v=\frac{dr}{dt}=\lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t}$$

where $v_{\text{avg}}={\Delta r}/{\Delta t}$, is measured by a speedometer (tachymeter) of a car. It is correct to affirm that: the velocity measured by a tachymeter is the instantaneous velocity?

For my humble opinion this affirmation is a false because not exists a precise instrument, that is always subject to errors and it not can never measure an instantaneous velocity.

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    $\begingroup$ Well any instrument will have some error, so I am not sure what you are really asking here. This is like saying a meter stick doesn't actually measure length because you don't have an infinite number of tick marks on the meter stick and you don't have infinitely precise vision. $\endgroup$ – BioPhysicist Nov 18 '19 at 21:38
  • $\begingroup$ @AaronStevens I want to stop the closure of my question, and therefore I have edited it. Is it clearer now? $\endgroup$ – Sebastiano Nov 18 '19 at 21:45
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    $\begingroup$ I suppose so. Although I still don't understand your reasoning. $\endgroup$ – BioPhysicist Nov 18 '19 at 21:56
  • $\begingroup$ @AaronStevens I simply wanted to know if what that is written on the textbooks is correct, that is, that a tachymeter of a car measures the instantaneous velocity. $\endgroup$ – Sebastiano Nov 18 '19 at 22:09
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    $\begingroup$ First you say that the average velocity is measured by a speedometer. Then you say that a speedometer measures instantaneous velocity. That makes no sense. $\endgroup$ – G. Smith Nov 19 '19 at 0:27
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For all practical purposes, it is the instantaneous velocity. All tools have an error. Let me pose a question: if you see an object move, do you see its instantaneous velocity at the current moment or its velocity a fraction of a millisecond ago, since it takes time for the light to hit your eyes?

Any tool for these type of measurements works by taking the average in a small fraction of time. This explains the $\Delta t \to 0$, as opposed to $\Delta t = 0$. There is nothing that will measure over a period of no time - then the expression would be irrational, dividing by $0$. But $\Delta t$ does approach $0$, though it never reaches it. This means that for all purposes, it is instantenous - this is the power of calculus.

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  • $\begingroup$ I always give credit for the effort of a well-prepared question. I have not understood (for my constant tired) this affirmation: if you see an object move, do you see its instantaneous velocity at the current moment or its velocity a fraction of a millisecond ago, since it takes time for the light to hit your eyes? I think that the instantaneous velocity is the photo for an instant fixed $t_0$. Does a tachymeter definitely measure average or instantaneous velocity? $\endgroup$ – Sebastiano Nov 18 '19 at 21:54
  • $\begingroup$ @Alaz Using your logic, which I agree with, it would also be accurate (though pedantic) to say that "there is no such thing as instantaneous velocity." Not sure relevant to the question though, just my own musings. $\endgroup$ – BobtheMagicMoose Nov 18 '19 at 21:57
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    $\begingroup$ @Sebastiano looking at your definition, $\Delta t$ is approaching $0$, which is what the tool gives. It is not exactly $0$ - an approximation only but a strong one. For a tool to display the instantaneous velocity at the same exact point in time as it happens is impossible. So, technically it is not instantaneous but it is close and we take it to be as such. As for your "photo", it displays the idea of $\Delta t =0$ well: there is no movement in the picture, and thus you cannot measure the velocity. $\endgroup$ – Alaz Cig Nov 18 '19 at 22:06
  • $\begingroup$ @BobtheMagicMoose not in the practical world. If there was a hypothetical body following a displacement over a period of time which is represented perfectly by a continuous function of time, taking the derivative with respect to time satisfies the defined requirement. Since the derivative is defined by approaching $0$, it solves this perfectly. $\endgroup$ – Alaz Cig Nov 18 '19 at 22:09
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    $\begingroup$ Just one funny anecdote: Isaac Newton has said, that he is a pure mathematician (mathematician, that only deals with pure mathematics (en.wikipedia.org/wiki/Pure_mathematics)), but nowadays we use his idea of calculus in physics. $\endgroup$ – User123 Dec 22 '19 at 20:39
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The strict answer is that the speedometers on modern cars show a rolling average gauged over a short time period, and not a truly instantaneous speed. The display is determined by the rate at which sensors pick up the rotary motion of the car's wheels, which is done on a sampling basis, with a definite time lag (albeit a short one) between the samples. Usually the display includes a damping factor to prevent rapid small changes to the displayed speed.

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