# Conservation of momentum in inelastic collisions

if energy is not conserved in inelastic collisions(eg two ball collision), the energy can be lost as heat or sound, but won't the sound or heat cause the air molecules to move so the molecules also acquire momentum? Then how can momentum be conserved? or do the net momentum of the two balls system decrease?

• Expand your definition of the system. Apr 19 '17 at 19:12
• exclude the air, just the two balls Apr 19 '17 at 19:29
• " but won't the sound or heat cause the air molecules to move so the molecules also acquire momentum?" - How can you say you are excluding the air if you are taking it into account? Apr 19 '17 at 19:32
• uhm, I think what I meant is: there are energy lost out of the system(two ball) as heat, wouldn't the momentum lost to outside the system as well as the energy causes the objects outside the two ball system to move Apr 19 '17 at 19:36
• Which objects outside the two-ball system? Apr 19 '17 at 20:10

It is a question of scale.
If the linear momentum changes during a collision due to the production of sound that change in linear momentum will be very much smaller than the total linear momentum of the colliding bodies.

In an inelastic collision the kinetic energy decreases due to the work done to permanently deform the colliding bodies and in the generation of heat and sound.

The sound waves will be generated in all directions so the net momentum change to the colliding bodies would be very small because the net linear momentum of the sound waves is so small.
The heat and permanent deformation is generated by internal processes involving internal forces within the colliding bodies and so the net linear momentum of the bodies will not change.
Just because bits of a body are moving around internally does not matter as it is the motion of the centre of mass of the body which is important.

Microscopically, energy and momentum is conserved even at inelastic collisions. In the energy conservation law you have to incorporate potential energies $V$ due to the binding forces inside the colliding bodies. This potential energy is given by the sum over all single intermolecular/interatomic interactions.

In quantum mechanics, you have also the conservation of energy and momentum up to an uncertainty $\Delta p \geq \frac{\hbar}{2 \Delta x}, \Delta E \geq \frac{\hbar}{2 \Delta t}$. This uncertainty becomes relevant on very tiny length and space scales $\Delta x, \Delta t$.

• thank you for the reply, I'm actually not at the level of understanding any quantum mechanics yet. So does the momentum of the two balls got lost to the surrounding? Apr 19 '17 at 19:32
• Yes, it gets lost to the surrounding Apr 19 '17 at 19:33
• okay thank you, so bascially I am still in high school, so if a question goes like this: a 5kg ball with 2ms-1 strikes an 10 kg ball at rest, and the collision is inelastic, after the collision, the total momentum of the 5kg ball and 10 kg ball will be slightly less than 10 Ns? Apr 19 '17 at 19:41