2.1 Conservation of Momentum

Consider two objects interacting with each other via some forces. They could be two electrons repelling each other because of the electrostatic force or two planets falling together due to gravity. By Newton’s third law,

\boldsymbol{F}_{A\to B}(t)=-\boldsymbol{F}_{B\to A}(t),

(2.1)

and so by Newton’s second law we can write

m_{A}\boldsymbol{a}_{A}(t)+m_{B}\boldsymbol{a}_{B}(t)=0.

(2.2)

We can integrate this equation over some arbitrary time period $t_{1}<t_{2}$ to get

$\displaystyle\int_{t_{1}}^{t_{2}}\left(m_{A}\boldsymbol{a}_{A}(t)+m_{B}% \boldsymbol{a}_{B}(t)\right)\operatorname{\mathrm{d}}\!t$	$\displaystyle=m_{A}\left(\boldsymbol{v}_{A}(t_{2})-\boldsymbol{v}_{A}(t_{1})% \right)+m_{B}\left(\boldsymbol{v}_{B}(t_{2})-\boldsymbol{v}_{B}(t_{1})\right)$	(2.3)
	$\displaystyle=0$	(2.4)
$\displaystyle\implies m_{A}\boldsymbol{v}_{A}(t_{1})+m_{B}\boldsymbol{v}_{B}(t% _{1})$	$\displaystyle=m_{A}\boldsymbol{v}_{A}(t_{2})+m_{B}\boldsymbol{v}_{B}(t_{2}).$	(2.5)

Thus, we have discovered that Newton’s third law implies that the quantity $m_{A}\boldsymbol{v}_{A}+m_{B}\boldsymbol{v}_{B}$ is conserved. This means it is constant for all time. We call this quantity the linear momentum.