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