Physics Study Guide/Print version - Wikibooks, open books for an open world

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Conservation of momentum

$\mathbf{p}_i = \mathbf{p}_f$

$\vec{L}_i = \vec{L}_f$

Let us prove this law.

We'll take two particles, say, a and b. Their momentums are $\vec p_a$ and $\vec p_b$ .They are moving opposite to each other along the x-axis and they collide. Now force is given by:

$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$

According to Newton's third law,the forces on each particle are equal and opposite.So,

$\frac{\mathrm{d}\vec{p}_a}{\mathrm{d}t} =- \frac{\mathrm{d} \vec{p}_b}{\mathrm{d}t}$

Rearranging,

$\frac{d (\vec{p}_a + \vec{p}_b)}{\mathrm{d}t} =0$

This means that the sum of the momentums does not change with time. Therefore, the law is proved.