# Question #f46fd

Feb 25, 2018

The principle of conservation of momentum

#### Explanation:

Newton's third law, namely that every action has an equal and opposite reaction

${F}_{1} = - {F}_{2}$

is really a special case of the conservation of momentum.

That is, if the total momentum in a system must be conserved, the sum of the external forces acting on that system must also be zero.

For instance, if two bodies collide with one another, they must produce equal and opposite changes in momentum in one another for the total momentum in a system to remain unchanged. That means, they must also exert equal and opposite forces on one another.

Here's the maths to go with it:

1) ${F}_{1} = - {F}_{2}$

2) Since $F = m a$

${m}_{1} {a}_{1} = - {m}_{2} {a}_{2}$

3) Since $a = \frac{\delta v}{\delta t}$

$\frac{\delta m {v}_{1}}{\delta t} = - \frac{\delta m {v}_{2}}{\delta t}$

$\frac{\delta {p}_{1}}{\delta t} = - \frac{\delta {p}_{2}}{\delta t}$

And finally, through some integration with respect to $\delta t$,

${p}_{1} = - {p}_{2}$