Conservation of linear momentum with no rest?

Post Reply New Topic

The law of conservation of linear momentum predicts that objects at rest stay at rest and objects in linear movement stays in movement without any speed derivate.

It made me think that it's somehow is an anomality that objects at rest would be a special case, and be asymetrical.

Could it be that ALL objects are in linear motion and the only characteristics that change is speed. Such that there's no "rest" only different speed relative to other objects?

You mean Newton's 1st law: Fnet=0, a=0.
No force= no acceleration= you stay moving at constant speed (0 included)

I don't understand "asymetrical" in this case? What did you mean there?


Objects ARE in linear motion pretty much, just in different directions at the same time- right and up for example.

Even a circle motion like a planet or satelite is linear if you take it over a very small time period. A circle is just a billion lines!


Right now we are spinning around earth's axis, while orbiting the sun, while the galaxies spin around. It's all relative to some point of reference. We just take it from where we can see 'cause it simplifies the math. Just a bit anyways!

The conservation of momentum is embodied in Newton's Third law, not in the First or Second laws.

See http://en.wikipedia.org/wiki/Newton's_laws_of_motion

Given a Force exerted by body 1 on body 2 there is and equal opposite force exerted by body 2 on body 1.

Let the first force be F12 and the second F21. Now F12 dp1/dt where p1 is the momentum of the first body at the time of contact. F21= dp2/dt where p2 is the momentum of the second by at the time of contact. F12 = -F21 therefore dp1/dt = - dp2/dt . Rewrite this as dp1/dt + dp2/dt = 0. Then by the distributive law d(p1 +p2)/dt = 0 which means the total momentum of the two bodies is constant which is what the law of conservation of momentum says. QED.

ruveyn

You are correct. Eisenstein also thought that the notion of a special "at rest" frame of reference was absurd. That's why he developed the special theory of relativity. According to the special theory of relativity, there is no unique "rest" frame of reference. Any inertial frame of reference is equally valid in terms of the governing laws of motion. The Newtonian laws are only an approximation for relatively slow moving objects though, as is the linear transformation used to transform your coordinate system from one frame of reference to another. According to relativistic calculations, the linear transformation is an approximation that only works when object speeds are much slower than the speed of light. This is because whenever you speed up your frame of reference time slows down in that frame by a small degree. This might seem absurd but the laws ultimately describe the same processes regardless of how time may expand or contract in different reference frames.

Seems I was right then

What made me think on this was that to change the speed of an object requires energy transfer, be it negative or positive. Now that would mean that an object at rest somehow got speed introduced.. but no energy. Probably formally wrong but it's at some level an anomaly that doesn't fit in with the rest.

The amount of kinetic energy attributed to a mass can only be defined relative to some inertial reference frame. This is true for both Newtonian and Relativistic mechanics. The important property is that the total energy of the system is conserved over time.

For example consider two bouncy balls of mass 1 kg, each moving towards the other at a velocity of 1 m/s. Then with a perfectly elastic collision the two balls simply trade their kinetic energies. After the collision each ball's momentum has reversed directions but the energy of each remains 1 joule. However, if you have the frame of reference move in the direction of the initial velocity of the left ball the same event will appear as the right ball moving to the left at 2 m/s and colliding with the stationary left ball. The right ball becomes stationary as it has transferred all of its kinetic energy (4 joules) to the left ball. If your frame of reference moves in the direction of the initial velocity of the right ball the energy will be transferred from left-to-right instead of right-to-left as in the previous case. The bottom line is the definition of kinetic energy depends on a specific choice of inertial frame of reference, but the total kinetic energy of the system is conserved through time in all cases. I realize this would be much easier to explain with a moving picture, ugh.

There are frame independent types of energy too though. The kinetic energy of a photon which always moves at the speed of light and the "rest energy" of a mass which is always given by E = m*c^2 (where m is the resting mass and c is the speed of light) are two types of energy that don't depend on any particular frame of reference.

As an aside, note there are also inelastic collisions in which momentum is still conserved but a certain amount of kinetic energy will be converted into internal thermal/molecular energy during the collision.

Last edited by marshall on 27 Oct 2012, 3:12 pm, edited 3 times in total.

I dont understand what it is that you dont understand.

How did your "object at rest" - "get speed introduced"?

It doesnt have any speed. Its AT REST.

Thats the whole damned point.

If a ball is moving at zero miles per hour- its "an object at rest".

A second ball moving in a certain direction through outer space at 100 miles an hour is "an object in motion".

If you give either ball a little push both will be effected.

The first ball would accelerate from zero to (say) five miles an hour.
At that moment it would graduate from being an 'object at rest' into also being 'an object in motion' along with the other ball.

The second (assuming that you nudge it in the same direction that its already moving) will accelerate from 100 to 105 mph.

Both balls respond the same way. They both accelerate in response to your energy input.

So how exactly is the ball that was at rest behaving in a way that is "a special case'?

The point is that it's an illusion that any object is at "rest". One could say the object have the same speed vector as the observer. The core of it is that energy causes changes to preexisting speed vector in all objects.

What does this have to do with your OP?

The first sentence in this post is obvious.

Obviously nothing in the universe is really stationary, or at "rest".
And all motion, and all non motion, is relative. They knew that long before Einstein.

The second is exactly what Newton said. It takes energy to change velocity, and that it doesnt matter whether the velocity is zero or nonzero.

But for some reason you objected to that statement by newton in your first post-but now you're spouting the same statement.

In the first post you complained that Newton saying "an object at rest will remain at rest" is somehow "asymetrical" to him saying "an object at motion will remain in motion". Its an example of symetry, and not asymetry. If you're going zero mph it takes as much energy to change that velocity as it does to change a 100 mph velocity. How is that asymetrical?

You can nitpick and take out the phrase "at rest" and replace it with "moving at the same velocity as the observer", but what Newton said about said objects remains the same.

The object is not going to change its vector relative to you without a force acted upon it whether its in a paralell vector as you or not.

There is no true rest, but "rest" (choosing a frame of reference which makes sense) makes the math easier (read: possible.)

I think you are just confused by bad wording.

Conservation of Linear momentum is essentially a symmetry of physical reality. If the entire cosmos had come into being just ten miles north of where it is now, the laws of physics would still be the same. This rather fanciful way of talking the the essence of Noether's Theorem which links the underlying symmetries of nature to her conservation laws.

Look up Noether's Theorem. It is very deep and very important.

ruveyn

The point is that you are wrong.

If you are in a vehicle that is moving at constant velocity in a straight line and the vehicle is enclosed so you can't look outside, there is no experiment you can do in the vehicle that will tell you that it is moving. Galileo made this point in Dialogs on Two New Sciences. Newton claimed that absolute time and rest existed but could not be detected by any empirical means.

ruveyn