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Conservation of Linear Momentum
Consider a 1-dimensional collision between two bodies in which no external forces act.
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 of motion states that FBA = - FAB
If the bodies are in contact for a short time, t, then
change in momentum of A is pA = FBA × t = - FAB × t
change in momentum of B is pB = FAB × t
So the total change in momentum (pA + pB) is zero.
The principle of conservation of momentum is, in effect, an expression of Newton’s third law 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). Calculate the magnitudes and senses of the velocities of the bodies after the collision.
Mass of A = mA = 2 kg
Mass of B = mB = 3 kg
The principle of conservation of momentum can be used
As the collision is elastic we can use the fact that the total K.E. is the same before and after the collision.
This gives us:
and these two equations can be used to calculate vA and vB.
However, the calculation can be simplified if we remember that for an elastic collision, the relative velocity of approach is equal to the relative velocity of separation.
velocity of A relative to B before collision is
velocity of A relative to B after collision is -2 ms-1
using this and equation 1 allows us to calculate vA and vB easily.