Show that Newton's third law of motion is contained in the second… - Vidyadip

Question

Show that Newton's third law of motion is contained in the second law.

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Answer

Let $F _{ ba }$ be the force (action) exerted by A on B and $\frac{d p _B}{d t}$ be the resulting change of the momentum of B.
Let $F _{ AB }$ be the force (reaction) exerted by B on A and $\frac{d p _A}{d t}$ be the resulting change of momentum of A.
According to Newton's second law, $F =\frac{d p }{d t}$
Then, $F _{B A}=\frac{d p _B}{d t}$ and $F _{A B}=\frac{d p _A}{d t}$
$
\therefore \quad F _{B A}+ F _{A B}=\frac{d p _B}{d t}+\frac{d p _A}{d t}=\frac{d}{d t}\left( p _B+ p _A\right)\ldots(i)
$
In the absence of any external force, the rate of change of momentum of the whole system zero.
i.e. $\frac{d}{d t}\left( p _B+ p _A\right)=0$
So, $F _{ BA }+ F _{ AB }=0$ or $F _{ BA }=- F _{ AB }$
or Action $=-$ reaction
and it is a Newton's third law of motion.
Hence, proved.

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