Without using law of conservation of linear momentum, prove newton's tird law ON NUMERICAL OR THEORETICAL BASIS(NOT EXPERIMENTAL BASIS i.e. based on real life examples).

Consider two bodies of mass m₁ and m₂

 

Their total momentum is P = P₁ + P₂

 

Now, the net force from Newton's second law is given as

 

F = dP/dt = dP₁/dt + dP₂/dt

 

Now, at the time of contact (this is needed so that third law becomes applicable) no net force acts on the system.

 

Thus,

dP₁/dt + dP₂/dt = 0

 

f1 + f2 = 0

 

Thus, f₁ = -f₂

 

This is Newton's third law.