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Conservation of Momentum and Newton's Laws of Motion
Consider a 1-dimensional collision between two bodies in which no external forces act, as shown below.

FAB is the force exerted by body A on body B.
FBA is the force exerted by body B on body A.
Newton's third law states that

Suppose that the bodies remain in contact for a short time ∆t, then the change in momentum of A is given by

and the change in momentum of B is given by

 so the total change in momentum is clearly equal to zero

This shows that the principle of conservation of linear momentum is an inevitable consequence of Newton's second and third laws of motion.

Elastic collisions: An Example
Two bodies A and B, have an elastic collision during which no external forces act (this can easily be arranged by having magnets with similar poles facing each other, as shown below).

Mass of A = mA = 2kg
Mass of B = mB = 3kg
Calculate the magnitudes and senses of the velocities of the bodies after the collision.

Assuming there are no external forces, the principle of conservation of momentum can be used, so we have

which, putting in the values for masses and velocities before collision, gives us
Equation 1
As the collision is elastic we can use the fact that the total K.E. is the same before and after the collision.
This can be expressed as follows
Equation 2
These two equations could be used to find the values of vA and vB however, the calculations can be simplified if we remember that the relative velocity of approach is equal to the relative velocity of separation, for an elastic collision (see here for proof).

We can therefore use the following equation (in combination with Equation 1 above) to find our answers

From the information given near the diagram we can see that

therefore

Combining this last equation with Equation 1 above allows us to calculate vA and vB easily.

The results are: vA = 1.6ms-1 and vB = 3.6ms-1
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