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# Prove Newton's first and third law of motion using Newton's second law of motion

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## Newton's second law of motion is F = ma. That is, we have:- F = m ( v-u / t ) Thus, Ft = mv - mu. Now, when F = 0, then v = u. That is, in the absence of force, the object continues to move with same velocity throughout. Now, when F = 0 and u = 0, then v = 0. That is, an object at rest will remain at rest if no force is acting on it. Thus, Newton's first law is derived from second law. Now 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 & F21be 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 = d(p2+p1)/dt We know that, no force is applied. Thus, momentum change will also be 0 because no change in velocity occurs. Thus d(p1 + p2)/dt = 0 Therefore, F12 + F21 = 0 That is, F12 = - F21. Thus, Newton's third law is proved with Newton's second law.

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