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The following question on Philosophy SE https://philosophy.stackexchange.com/q/73366/ relies on this "given"

"suppose that the instantaneous velocity of object A is $1$m/s and that the mean velocity of object B is also $1$ m/s..." .

I'm not interested here in the question that is asked under this assumption, but in the assumption itself.

Is it the case that the same number with the same unit can be both the value of an instantaneous velocity and the value of a mean velocity?

My hypothesis is that the unit of instantaneous velocity is not really m/s for a the second is a finite lapse of time, while instantaneous velocity is a derivative, hence a limit, namely the limit of the ratio of change in distance over an infinitely small lapse of time :

lim$_{\Delta(t)\rightarrow 0} \frac {\Delta(distance)}{\Delta(t)}$

From a purely mathematcal standpoint is it correct to say that the unit of instantaneous velocity is m/s? Or, put differently, does "m/s" mean the same thing when used for instantaneous velocity and for mean velocity?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – David Z
    Jun 2, 2020 at 21:47
  • $\begingroup$ The units don't change just because you are taking a limit. $\endgroup$ Jun 3, 2020 at 2:39

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Is this a correct interpretation of your question:

  • If I speed up from $0 \,\mathrm{km/h}$ to $100 \,\mathrm{km/h}$ in 10 seconds, then it might sound odd to say that I drove $50 \,\mathrm{km/h}$ at some point. Obviously I never actually drove for one hour, so how does 50-kilometres-per-hour make sense? Such "non-instantaneous" unit can't describe an instantaneous event.

If this is correctly understood, then good point. I get the confusion. Your claim is that such a $ \,\mathrm{km/h}$ unit should not apply to instantaneous, but only to non-instantaneous, finite values.

The problem is just that this logic also fails with averages / means:

  • Say, you drive with exactly $50 \,\mathrm{km/h}$ from you enter the motorway till you leave it again. So, your average / mean speed is $50 \,\mathrm{km/h}$. Now, this may only last 10 minutes, before you are off the motorway. How does it make sense to say 50-kilometres-per-hour when you drove for only 10 minutes? Apparently, such $ \,\mathrm{km/h}$-unit also cannot apply for averages/means...

The point is that both of these unit interpretations are wrong. The correct interpretation of any unit works for both cases (for any cases). And that correct interpretation or understanding is:

The km/h-unit tells how far you will move if you kept that speed for one full hour.

The "if" is the key. Sure, you don't keep that speed for one full hour. But if you did, then that would be the distance you traveled. Thinking in this way every time we see this unit makes all values comparable. Which after all is the very purpose of units.

(Now, replace $ \,\mathrm{km/h}$ with $ \,\mathrm{m/s}$ and repeat this logic, and you'll get to the same conclusion.)

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    $\begingroup$ +1: Feynman discusses this at some length in Section $8.2$. $\endgroup$
    – user87745
    Jun 2, 2020 at 23:40
  • $\begingroup$ Thanks for the reference. $\endgroup$
    – user232501
    Jun 3, 2020 at 8:04
  • $\begingroup$ Just to make sure I understood correctly. Saying that an object has a 1 m/s instantaneous velocity means that, if continued its movement without being accelerated ,then , after one second, it would have covered a 1 meter distance. $\endgroup$
    – user232501
    Jun 3, 2020 at 8:27
  • $\begingroup$ Is it correct to say that instantaneous velocity is conditional mean velocity ( the condition being , " if the object continued its movement without being accelerated for , say, 1 second, or, say, 1 hour", depending on the context). $\endgroup$
    – user232501
    Jun 3, 2020 at 8:30
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    $\begingroup$ @RayLittleRock But should mean velocity also be considered a conditional velocity then? The condition should be the exact same, namely that the object continues its motion unaccelerated for the duration. I guess you can think about it as you do - but I do not follow, why you are trying to make instantaneous velocity into conditional mean velocity, when mean velocity itself would adhere to the same condition. $\endgroup$
    – Steeven
    Jun 3, 2020 at 8:32

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