Does Instantaneous Velocity Still Make Sense in Planck-Quantised Time and Space?

[EngTechno]

I know nothing about Instantaneous Velocity. Can you give me the very simple form of explanation? Is instantaneous velocity an exact velocity at an exact point?

 

[mathman]

Is instantaneous velocity an exact velocity at an exact point?

Yes. One way of approaching it is through elementary calculus. Consider a small interval around the point of interest, and divide the interval by the time it takes to cross it. This is the average velocity. The limit as the interval goes to zero is the instantaneous velocity. If you use the length of the interval, you get speed.

 

[HallsofIvy]

That was, in fact, the impetus for the creation of calculus. We know that F= ma but if gravitational force depends upon distance, then we should be able to calculate the force at that instant- but ma is not defined at a distance, since acceleration (change in velocity) requires a time to change! In other words, people trying to figure out what kept planets in their orbits had to come to the conclusion that "F= ma" made no sense! It required the concept of limits and the derivative to solve that problem.

 

[robphy]

Yes. One way of approaching it is through elementary calculus. Consider a small interval around the point of interest, and divide the interval by the time it takes to cross it. This is the average velocity. The limit as the interval goes to zero is the instantaneous velocity. If you use the length of the interval, you get speed.

[Instantaneous] Speed is the magnitude of the [instantaneous] velocity vector.

EngTechno,
Was there a problem with the answers provided here [thread]40372[/thread]?

 

[JonF]

Another way to think of instantaneous velocity the rate displacement is changing at a given instant.

 

[HallsofIvy]

Another way to think of instantaneous velocity the rate displacement is changing at a given instant.

Except that, strictly speaking, since "change" itself requires a time interval, nothing CAN change "at a given instant"! That's why you need to work with limits in order to define "change at a given instant".

 

[robphy]

This discussion reminds me of a logical problem I see with the textbook development of velocity.

It seems that most texts follow the scheme:
first, "average velocity"
then, "[instantaneous] velocity".

It's strange to me to define the "average of a quantity" before defining the actual quantity.

In addition, it seems strange to me that there is little discussion that one is really doing a time-weighted-average of velocity and not a straight-average of velocity.
For a piecewise constant-velocity trip,
$$v_{avg} \equiv \frac{\int v\ dt}{\int dt}=<br /> \frac{v_1\Delta t_1 + v_2\Delta t_2 + \cdots + v_n\Delta t_n}<br /> {\Delta t_1+\Delta t_2+\cdots+\Delta t_n}<br /> =<br /> \frac{\Delta x_1 + \Delta x_2 + \cdots + \Delta x_n}<br /> {\Delta t_1+\Delta t_2+\cdots+\Delta t_n}<br /> =\frac{\Delta x}{\Delta t}$$

 

[Ethereal]

This may sound a little naive, but if time and space were Planck-quantised, does it still make sense to speak of "instantaneous" velocity? The limit can't go to zero in this case.

 

[HallsofIvy]

This may sound a little naive, but if time and space were Planck-quantised, does it still make sense to speak of "instantaneous" velocity? The limit can't go to zero in this case.

That's a physics question, not a math question!
It would still make sense to treat, in certain problems, velocity as distance and time interval were continuous.