# Conservation of Linear Momentum and Inelastic Collisons

## Main Question or Discussion Point

I don't understand how inelastic collisions still conserve momentum.

If kinetic energy is not conserved, velocity must change, and the mass obviously doesn't change, how can momentum be conserved? It makes no sense to me at all.

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During collisions (where mass is constant) velocity generally changes .However it changes such that momentum is conserved i.e if one body gains momemtum(increase in velocity along initial direction) the other loses momentum(loss in velocity along initial direction)

So mu+MV is constant where m, M are mass and u and V are velocity vectors at any instance.

During collision u and V can change however the sum mu+MV muat remain constant.

This in no way implies that mu^2/2 +MV*2/2 is constant.

So sum of kinetic energy may or may not remain constant.

For eg: take m =1 M =5
And u1 =3 and V1= 4 (all in SI units)
after collision suppose u2=-2
So V2 has to become 5.

So not only is individual kinetic energy of bodies changing after collisions, the sum of kinetic energies is also changing.

Nugatory
Mentor
I don't understand how inelastic collisions still conserve momentum.

If kinetic energy is not conserved, velocity must change, and the mass obviously doesn't change, how can momentum be conserved? It makes no sense to me at all.
Suppose I have two objects, same mass, both at rest. Momentum and kinetic energy is zero.

Now consider the same two objects, but one is moving left at speed u, the other is moving right at speed u. Momentum is still zero (mu + -mu = 0) but the kinetic energy is not zero.

If they collide head-on and stick together (completely inelastic collision) they end up both at rest. Momentum is conserved because it's zero either way. Kinetic energy isn't conserved, but the total energy is conserved; all the pre-collision kinetic energy has turned into heat.

Momentum is conserved because of Newton's law, every force has an equal and opposite force. If one object pushes the other for some time with an average force of F, it feel on itself in the other direction the same force F for the same time. Since force is proportional to mass and acceleration, if one is twice the mass of the other, it feels half the acceleration, four times then one fourth etc. Since the acceleration lasts the same time, the velocity times mass stays the same if you consider both of them (eg a five times heavier object's velocity changes five times less, while a five times lighter objects velocity changes five times more, if you multiply momentums together and add them up, they stay the same before and after the collision).

The energy might not be conserved, because as one object collides into the other, it causes deformation and depending on whether the deformation pushes back or stays deformed (increasing heat) the kinetic energy is conserved or not. The momentum is still conserved, because the force needed to cause the deformation is still felt by both of the objects for the same time, but if the deformation is not elastic it uses up some of the potential to do work.