Question

Derive the law of conservation of momentum from Newton's third law of motion.

Answer 1

Derive the law of conservation of momentum using Newton's third law

1. We know that Newton's third law states that when two bodies interact, they apply forces to one another that are equal in magnitude and opposite in direction. The law of action and response is another name for the third law.
2. If consider a system of two bodies of mass $m_1$and $m_2$ And if $\overrightarrow{F_{12}}$and $\overrightarrow{F_{21}}$are the forces exerted by the mass $m_1$ on $m_2$and mass $m_2$and $m_1$respectively.
3. So, hereby third law, we can write, $\overrightarrow{F_{12}}$= $-\overrightarrow{F_{21}}$ Then, $\overrightarrow{F_{12}}$ + $\overrightarrow{F_{21}}$=0
4. Then we can write, $\overrightarrow{F_{12}}=\frac{d\overrightarrow{p_1}}{dt}$and $\overrightarrow{F_{21}}=\frac{d\overrightarrow{p_2}}{dt}$
5. Then, $\overrightarrow{F_{12}}$+$\overrightarrow{F_{21}}$=$\frac{d\overrightarrow{p_1}}{dt}+\frac{d\overrightarrow{p_2}}{dt}$
6. Then, $\frac{d(\overrightarrow{p_1}+\overrightarrow{p_2})}{dt}=\frac{d\overrightarrow{p}}{dt}$Here p is the total momentum of the system.
7. Now if net external force is zero, $\overrightarrow{F_{12}}$ + $\overrightarrow{F_{21}}$ = $\frac{d(\overrightarrow{p_1}+\overrightarrow{p_2})}{dt}=\frac{d\overrightarrow{p}}{dt}=0$
8. Then,$\frac{d\overrightarrow{p}}{dt}=0$

Hence, from this equation, we can say that the total momentum of the system remains constant if the net external force is zero.