Question

Can you arrive at Newton's 3rd law using Newton's 2nd Law ?

Answer 1

Derivation of Newton’s third law of motion from Newton’s second law of motion
Consider an isolated system of two bodies A & B mutually interacting with each other, provided there is no external force acting on the system.

Let FAB, be the force exerted on body B by body A and FBA be the force exerted by body B on A.

Suppose that due to these forces FAB and FBA, dp1/dt and dp2/dt be the rate of the change of momentum of these bodies respectively.

Then, FBA = d p1 /dt ---------- (i)
=> FAB = d p2 /dt ---------- (ii)

Adding equations (i) and (ii), we get,
FBA + FAB = d p1 /dt + d p2/ dt

⇒ FBA + FAB = d( p 1 + p 2 ) /dt

If no external force acts on the system, then

d( p 1 + p 2 )/ dt = 0

⇒ FBA + FAB = 0
⇒ FBA = - FAB---------- (iii)

the above equation (iii) represents the Newton's third law of motion (i.e., for every action there is equal and opposite reaction)