Question

Can we obtain Newton’s 3rd law using Newton’s 2nd law?

Answer 1

Let us consider a system of 2 bodies 1 & 2 and consider that there is no external force acting. Now, let F₁₂ be the force acting on 2 by 1 & F₂₁ be the force acting on 1 by 2.

The rate of change of momentum of 1 = dp₁/dt and rate of change of momentum of 2 = dp₂/dt.

Thus, according to Newton's second law of motion F₁₂= dp₂/dt and F₂₁= dp₁/dt

Adding both the above equations, we get

F₁₂+ F₂₁= dp₂/dt + dp₁/dt

We know that, no force is applied. Thus, momentum change will also be 0 because no change in velocity occurs.

Thus, dp2/dt + dp₁/dt = 0

Therefore, F₁₂+ F₂₁= 0

That is, F₁₂= - F₂₁.

Thus, Newton's third law is proved with Newton's second law.