Question

obtain newtons 3rd law from newtons 2nd law

Answer 1

Third law of Newton was :
"To every action, there is an equal and opposite reaction and they act on two different bodies" .

Answer 2

According to Newton's Second Law, the rate of change of momentum is directly proportional to the mass of the body and inversely proportional to the force acting on it.

According to Newton's Third Law, every action has an equal and opposite reaction.

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 p 1 dt     ---------- (i)       => FAB =   d p 2 dt     ---------- (ii) Adding equations (i) and (ii), we get,  FBA+ FAB =   d p 1 dt +   d p 2 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).

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