# Momentum and K.E.

How can momentum always be conserved while kinetic energy is lost when the collision is inelastic? Since there is a loss of energy why does momentum before collision is still equal to momentum after collision?
Thanks

## Answers and Replies

I don't think there is a "why" answer to this question. The fact that (mechanical) energy is not conserved in an inelastic collision comes out of the requirement that the two bodies stick together. If you set up the equations and require momentum to be constant, but the masses combine into a single mass, energy cannot be conserved.

Andrew Mason
Homework Helper
How can momentum always be conserved while kinetic energy is lost when the collision is inelastic? Since there is a loss of energy why does momentum before collision is still equal to momentum after collision?
Thanks
This may be easier to see by looking at a collision between two bodies in the frame of reference of the centre of mass of the colliding bodies. Since the only forces are between the two bodies, there are no external forces acting on the two-body system so the motion of the centre of mass cannot change (by the first law).

But when the two bodies stick together after colliding, all motion with respect to the centre of mass ends. So kinetic energy in the centre of mass frame disappears.

BTW, it is not that the energy disappears. It can't. It is just that it is no longer macroscopic kinetic energy. The energy is converted to other kinds of energy.

AM

A.T.
Since there is a loss of energy
There is no loss of energy. There is just loss of macroscopic kinetic energy, which is converted into microscopic kinetic energy (heat). There is only one form of momentum, while energy can be converted into other forms, so the conservation of energy is less obvious.

Thanks. Now i understand ;)

Another way of looking at this (besides CoM) which I enjoy is to analyze the forces individually. By Newton's second law, F=dp/dt. Also, according to his 3rd law, the force exerted by mass 1 on mass 2 is equal and opposite to that exerted by mass 2 on 1. Let's say we have a two-body system. The total momentum of the system is given by the sum of the individual momenta of each mass, so p=p1+p2. If we take the time derivative: dp/dt=dp1/dt+dp2/dt. BUT what are dp1/dt and dp2/dt? They're just the forces on masses 1 and 2, respectively! And these forces must be equal and opposite, so dp1/dt=-dp2/dt (if the masses interact only with each other during a collision, or whatever their interaction may be). Then, we have that dp/dt=0. Or, momentum is conserved. However, there was no statement whatsoever here concerning energy. It was all just forces and momentum, and like the people above have mentioned, only macroscopic kinetic energy is lost, not actual energy. Sorry for the extra post, I know you understand it now. Just wanted to give a slightly different perspective :).

Khashishi
As an aside, energy and momentum are unified into the 4-momentum in special relativity, so energy conservation and momentum conservation are both one same law. You have to use the full energy though, and not just the kinetic energy.

Linear momentum is conserved as a result of newtons 3rd law?
I can think of one case where linear momentum is conserved while ‘linear ke’ isnt..
If two rough balls collide on a smooth surface.....they may gain angular momentum which adds up to be equal to that before the collision....however if they both spin faster, the would have more rotational kinetic energy.....but since total energy is conserved, the total linear ke would be reduced....

A.T.
Linear momentum is conserved as a result of newtons 3rd law?
Yes

I can think of one case where linear momentum is conserved while ‘linear ke’ isnt..
That is usually the case.

Andrew Mason
Homework Helper
Linear momentum is conserved as a result of newtons 3rd law?
Conservation of momentum is actually a bit more fundamental than Newton's third law. Newton's third law assumes absolute time and space, so it breaks down in Special Relativity and electromagnetism. But the law of conservation of momentum appears to have no exceptions.

AM

Conservation of momentum is actually a bit more fundamental than Newton's third law. Newton's third law assumes absolute time and space, so it breaks down in Special Relativity and electromagnetism. But the law of conservation of momentum appears to have no exceptions.

AM
I knew that the law of conservation of momentum applied at small scales and high speeds, but I thought Newton's 3rd law did as well! How does Newton's 3rd law not apply in some situations? Could you give me some links to sites/lectures/books that say this? Thanks :)

EDIT (clarification): I find this fact confusing because I always thought Newton's 3rd and momentum conservation were completely interchangeable concepts. Maybe you're thinking of Newton's 2nd?

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Andrew Mason
Homework Helper
I knew that the law of conservation of momentum applied at small scales and high speeds, but I thought Newton's 3rd law did as well! How does Newton's 3rd law not apply in some situations? Could you give me some links to sites/lectures/books that say this? Thanks :)

EDIT (clarification): I find this fact confusing because I always thought Newton's 3rd and momentum conservation were completely interchangeable concepts. Maybe you're thinking of Newton's 2nd?
See: http://en.wikipedia.org/wiki/Newton%27s_third_law#Importance_and_range_of_validity

See this attempt to get around the problem: http://en.wikipedia.org/wiki/Weber_electrodynamics#Newton.27s_third_law_in_Maxwell_and_Weber_electrodynamics

Also, have a look at this link: http://physics.stackexchange.com/questions/43269/violation-of-newtons-3rd-law-and-momentum-conservation

AM