Calculus & Motion: What Does Calculus Tell Us?

[JoeDawg]

What does calculus tell us about the nature of motion? If anything?
Or is it just an accurate way to calculate the effects of motion?

 

[petm1]

What does calculus tell us about the nature of motion?

That motion is the geometry of space-time.

 

[WaveJumper]

That motion is the geometry of space-time.

Did you mean "Motion is dependent/relative to the geometry of space-time"?

 

[mikelepore]

I don't know about the nature of motion; I think it tells us something about the nature of the way the brain produces intelligence. After we were unable to get an exact solution otherwise, we discovered that we could get an exact solution by visualizing a finite object or event as an infinitely large number of infinitely small parts. That's a fascinating characteristic of the human brain.

 

[petm1]

Did you mean "Motion is dependent/relative to the geometry of space-time"?

Geometry is the study of shape. Matter's intrinsic motion, we measure matter as a shape, tells space how to warp, how space is warped determines how matter moves through space. Calculus is the study of these changes, so imho motion is the geometry of space-time that calculus maps for us.

 

[Pythagorean]

What does calculus tell us about the nature of motion? If anything?
Or is it just an accurate way to calculate the effects of motion?

Motion requires two variables: space and time. We move a certain distance in a certain time, that is velocity. Because velocity is a rate (distance per time), we can plot the distance against the time, and look at the slope (the slope of a distance vs. time plot is equivalent to the velocity)

For the simplest case (constant velocity) the slope is a constant, and calculus isn't necissarily required. But in reality, things don't move at constant velocity, so the slope (distance/time) changes (i.e. velocity changes) as time changes.

Calculus is a way to frame this change as a function of the dependent variable (time, in this case). Since the slope is not constant, it will be an algebraic equation that depends on t, rather than a number that is constant for all t.

We can go further an plot the velocity as a function of time, once we have it's algebraic representation. Now the slope of this plot is the acceleration. Again, it can be constant (in which case simple algebra will tell you the acceleration) but more often it is not constant (so we must again call on Calculus to analyze the slope as a function of time and get a variable-based equation for the acceleration).

 

[techmologist]

It may be that calculus does not provide a description of all kinds of motion, but fortunately it does for most of the ones we encounter in everyday experience. Also, it might be an example of "looking for your keys where the lighting is good." Motions that are differentiable are easy (relatively speaking) to analyze, so we analyze them.

For instance, it seems to be the case that large-scale motions are approximately continuous with continuous first derivatives. That is, there are no infinite accelerations in the real world of large-scale things. For infinite accelerations you need an ideal world where there are, for example, perfectly rigid billiard balls that can experience an instantaneous transfer of momentum from one to the other. But it is possible in the real world to have discontinuity of acceleration. If you roll a flexible hoop down a circular ramp that straightens out at the bottom, the hoop will uncompress and hop off the ramp at the beginning of the flat part of the ramp, even though the ramp itself (and its slope) is continuous the whole way. So large-scale motions are well approximated by differentiable (and thus continuous) functions, but generally not by infinitely differentiable functions.

But as to whether continuity is an necessary property of motion, I don't know. It might be that it is only a good approximation over a very large and useful range of scales, just as the continuous medium approximation works well for fluid dynamics until you get down to the scale of individual molecules. If motion weren't at least approximately continuous, I don't know how we would ever establish that the thing that was here is identical to the thing that is now there. Experience would just be a blur of unconnected events.