Conservation of Momentum

Momentum is conserved in any collision if the effect of any external forces present is negliable relative to the effect of the collision. Consider a collision as shown in Figure (6.1).

 

Figure 6.1: 1-D Collision

Apply the impulse-momentum theorem to m₁ and m₂ separately,

$$\bar{F}_{1}^{} \Delta \Delta$$ =

$$\bar{F}_{2}^{} \Delta \Delta$$ =

where $\bar{F}_{1}^{}$ = the average force of m₂ on m₁ , and $\bar{F}_{2}^{}$ = the average force of m₁ on m₂ . By Newton's third law F₁(t) = - F₂(t) which gives $\bar{F}_{1}^{}$ = - $\bar{F}_{2}^{}$ and so,

$$\bar{F}_{1}^{} \bar{F}_{2}^{} \Delta$$

$$\Rightarrow$$ (4)

This is the statement of the conservation of momentum.

Note:

• The system must be isolated: the affect of all external forces acting on m₁ and m₂ must be negligable.
• The conservation of momentum holds for a collision involving any number of objects:
  

  $$\Sigma \vec{p}_{nf}^{} \Sigma \vec{p}_{ni}^{}$$ (5)

• Momentum is a vector, and each component is conserved separately. The equation for conservation of momentum really contains three equations, one for each dimension.