1. Conservation of Momentum in 1-D and collisions
Understanding Concepts

2. Law of Conservation of Linear Momentum
The law of conservation of momentum is defined as:
If all forces on a system of interacting particles are balanced, and remain that way, the linear momentum of the system before the interaction is equal to the linear momentum of the system after the interaction.

3. Conservation of Momentum Cont’d
To gain a better understanding of this law, lets go back to Newton’s third Law:
Now, using Newton’s 2nd Law, we obtain:
We can now use our definition of average acceleration to obtain:

4. Conservation of Momentum Cont’d
Because the time interval during which the forces act upon each other are equal, we have: or
However, we know that the change in momentum of an object is just it’s change in velocity:

5. Conservation of Momentum Cont’d
Now, doing a little algebra, it’s easy to show that:
Which simplifies to
Thus, we have the result that
This is the Law of Conservation expressed Mathematically

6. Collisions: Elastic and Inelastic
Now that we have discussed the law of conservation of momentum, it is now important to consider interaction types. Now because there isn’t only one way in which two or more systems can interact, or “collide”, and momentum is always conserved regardless of the interaction type, the conservation law has implications. But first, we must consider types of collisions that occur; namely “elastic” and “inelastic” collisions.
So, what’s the difference?

7. Collisions: Elastic and Inelastic
As we discussed briefly already in class, elasticity in a collision refers to how responsive the objects are involved to changes, or deformities, in their underlying atomic structure
Elastic: Total kinetic energy of the system is the same before and after the collision.
That is, the sum of the kinetic energies of all objects before the collision is equal to the sum of all objects afterwards
So, what does this tell us?

8. Collisions: Elastic and Inelastic
Atomic arrangements remain unchanged
Very little of the kinetic energy was transformed into electrical potential, or given converted into radiation
Inelastic: Total kinetic energy of the system before and after the collision is not conserved.
That is, the total kinetic energy before the collision is not the same afterwards.
What does this tell us?

9. Collisions: Elastic and Inelastic
The Atomic arrangements have changed
perfectly inelastic collisions are those in which the objects stick together during the collision
Electrical potential energy has been stored and/or used and radiated off as heat, or other forms of radiation
Conclusions
Momentum is always conserved
Elastic collisions conserve kinetic energy and atomic arrangement, inelastic collisions convert and/or use kinetic energy and change the atomic arrangement
Truly elastic collisions never really occur, accept between sub-atomic particles