Momentums and Curve Our Space is clearly a little weirder than expected

[Frogeyedpeas]

Newton's laws of motion state that objects in motion/rest will remain in motion/rest unless acted on by a force (yank, pull, jerk and any other such force/derivative of force) but then the question I beg to ask is why?

It makes sense that in an abstract empty block of space with no forces acted in it objects in a state of motion will continue to move in that state of motion but why does this property stop only at velocities and not apply to higher order derivatives of position?

Ex: an object in a state of acceleration with no net force acted on it. How is this possible?

When an object is in a state of motion (velocity) it has what we call momentum. Momentum is measured as velocity x mass and basically is the "quantity of motion" that allows object to move. No momentum, no velocity. Once you acquire momentum you acquire velocity. And a force is such a thing that gives

Likewise there exists a quantity called Second Order Momentum (we shall call it P2). P2 is defined as Acceleration x Mass (not to be confused with Net Force). It may be hard to notice P2 as of right now so here is a good abstract example.

Imagine a 5 kg rod of some extremely long length (say a lightyear) but rigid enough that it won't break should a force travel through it (diamond spandex?). If a force of 100 N is applied to the rod the rod will begin to accelerate... Now this force has to travel at the speed of sound (through the rod's medium) to reach to the other end (which will probably be take an upwards of a few years) but eventually if the force is continuously applied we will reach a time when the entire rod is in a state of acceleration which shold 20 m/s^2 (logically because 100/5 = 20). At this point suppose we were to remove the force from the end of the rod. Just as the rod did not begin accelerating instantaneously it will not cease accelerating instantaneously. Once again a wave moving at the speed of sound (through the rod's medium) has to travel from end to end before the entire rod stops accelerating. Until that period of time the rod is really accelerating by itself with no external forces acting (just a delayed internal force) so the quantity P2 (second order momentum) does exist in the rod for this period of time and is equal to 100 N but steadily decreasing.

Like that example after example can be made describing how a quantity P3,P4,P5, etc... can exist for a period of time but we still are left with a question. Only P1 (linear momentum) remains unchanged if an object in a state of motion is immediately isolated from all surrounding forces, yanks, etc... All the other higher order momentum(s) decrease if the object is immediately isolated.

Yet if we imagine us living in any sort of abstract empty space (whether curved or not to all you relativists) we see no reason for this. Clearly the object should just follow its normal free-fall trajectory (or the synonymous word Geodesic World-Line) with its current state of motion and should not at all be reducing in acceleration, jerk, change in jerk, change in change in jerk etc...

So even in terms of living in abstract empty space the Universe is not described properly and even with the notion of having curved space time (as is done in GR) does not account for the fact that this occurs. The question is why? And if there is anything closely related to it I might mention that as of right now there appears to be no upper bounds on acceleration or higher order derivatives of position, while velocity does have a maximum value: c. There is a connection there relativistically. But that does not explain this situation entirely. Yes it is true that objects in a state of acceleration should decrease as their velocity increases (due to Lorentz) but that does not answer why their should be an instantenous reduction in a state of acceleration in the absence of force.

Hopefully that all made sense,

Thanks for Reading and I would love to read your responses!

 

[Frogeyedpeas]

Just one thing, I used the rod example to simplify the problem a little, but as was made aware to me by another member of physics forums that isn't exactly how it works as the equation for motion will be a little more complicated.

 

[Vorde]

I'm confused a bit but I think it is within my ability to address the first part of your proposal.

Velocity is relative, an object can have any arbitrary velocity depending on the reference frame you are viewing it in. Instead of viewing Newton's first law as saying that an objects velocity will not change unless you apply a force to it, think of it as saying that if you do not apply a force to it, it will not experience any acceleration.

I'm not so sure about the 'quantity of motion' stuff, but I'm fairly sure "Once you acquire momentum you acquire velocity" is a bad way of looking at it. Momentum is not an intrinsic property, it is simply mass x velocity, the only thing special about this particular mathematical combination is the value of it in a system does not change over time.

The flaw in the rod example is that you are treating the rod as a single object where the crucial part of the argument is that different parts of the rod act in different ways.

I'm not quire sure what you are getting at in the rest of your post, but I believe I found your inconsistency. It doesn't matter that acceleration changes, acceleration isn't conserved (I'm not even sure what that would mean), momentum is.

 

[The_Duck]

Frogeyedpeas, regarding your rod example, no part of the rod ever accelerates without a force being applied. There is no such thing as acceleration sustained in the absence of a force.

Since the rod is a very extended object, and different parts of it accelerate differently, we should really consider what is going on in each part of the rod. It doesn't make sense to talk about "the acceleration of the rod" if different parts of the rod are accelerating differently. Therefore consider a small section somewhere in the middle of the rod. When you push on the rod, why does this section of the rod accelerate? After all, you are pushing on a different part of the rod entirely. The reason it accelerates is that a pressure gradient forms along the rod. The pressure near the pushed end is higher than the pressure near the far end. So the small rod section we are considering feels a larger pressure force on it from the pushed end than the far end, so there is a net force on this section of the rod, so it accelerates.

(The situation is identical to that of a fluid supporting itself against gravity, where any given small volume of fluid is supported against falling because the pressure below it is greater than the pressure above it, and this creates a net force on the volume which counteracts the force of gravity.)

Now, what happens when you stop pushing the pushed end? The pressure gradient starts to go away. But it does NOT vanish instantaneously along the whole length of the rod, rather the pressure gradient vanishes at a given point only when the sound wave produced by the end of the acceleration reaches that point. Until that sound wave gets to the section of the rod we were considering above, that section of the rod still experiences a pressure gradient, and thus a net force, and thus continues accelerating. This is just regular acceleration as produced by a force.