# Why is there no such thing as a body in a state of acceleration?

It appears that velocity is a quantity of motion meaning that all objects can have assigned to them a particular velocity. Through the application of forces (ex: gravity, E&m) we measure changes in velocity, which is called acceleration but here is the crux:

Once the force ceases to be applied the body will stop accelerating.... Meaning that while forces can be applied there is no actual quantity of force

There exists a quantity called momentum (mv) such that a body of mass m with momentum Q will move indefinitely velocity Q/m unless acted on by a force (m*a)

There theoretically could be this thing called 2nd momentum (ma) such that a body of mass m with 2nd momentum Q would accelerate indefinitely unless acted upon by a 2nd order force (mass*jerk)

The reason the above ^ does not occur is because for it to occur would require a constantly increasing amount of energy... ie infinite energy.

But we don't actually mean energy, we mean the quantity 1/2 * mv^2 would grow arbitrarily large for this to occur...

In an abstract mathematical world where it is not 1/2 mv^2 that is conserved but rather the quantity 1/2 m*a^2 that is conserved objects could be in a state of uniform acceleration indefinitely and still satisfy conservation of 2nd order momentum...

But obviously our universe IS NOT that abstract mathematical world of the 2nd kind... It is of the first... Why? What dictated that momentum (mv) became the quantity of motion and not (ma) or for that matter (mass *jerk) or why not fractional derivatives of velocity?

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Is this outside the realm of current research to determine? – frogeyedpeas May 4 '13 at 3:39
If you are familiar with Lagrangian mechanics, this question is actually equivalent to this one. That said, I'm disinclined to call it a duplicate, since they approach the issue from different directions. Also, it's not clear you'll get entirely satisfactory answers however this is asked. – Chris White May 5 '13 at 20:52
Thank you! This appears to be similar, it appears my actual physics knowledge versus what I want to know have a big disparity... Though I think this is a good step to fixing it – frogeyedpeas May 5 '13 at 22:54
I didn't realize this had been closed. Would someone enlighten me as to why a deep question about fundamental physical laws is "not constructive"? Perhaps it is a duplicate (see my above comment, and note that the answer there is not complete and seems to only apply to specific cases), but why is this perceived as argumentative? – Chris White Jul 20 '13 at 6:28

## 1 Answer

Your premise violates Newton's first law of motion:

If there is no net force on an object, then its velocity is constant. The object is either at rest (if its velocity is equal to zero), or it moves with constant speed in a single direction.

For an object (a body) to be accelerating there must be an external force applied. One of the reasons for Newton's first law is the conservation of energy and momentum. If an object could continue to accelerate without any external force applied it would be gaining energy relative to its environment.

In an abstract world where this was allowed (energy could come from nothing) none of the laws of physics that we know today would apply. I don't even think a Universe like that could exist.

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My question reduces to the following: why is momentum (mass * velocity) and energy (1/2 mass * velocity^2) conserved? Why isn't (mass * acceleration) and 1/2 ( mass * acceleration ^2) or some other derivative or combination of derivatives of motion – frogeyedpeas May 5 '13 at 22:58
@frogeyedpeas if momentum or energy were not conserved the universe would not be stable. Either they'd be lost due to interactions and the universe would die a slow cold death with no energy or momentum to do anything or, the universe would diverge into a hot explosive fireball of runaway energy and momentum. – Brandon Enright May 5 '13 at 23:17
@frogeyedpeas There are strong symmetry reasons for energy and momentum conservation. Look up Noether's theorem. – Emilio Pisanty May 12 '13 at 21:07