Question

Prove that Newton’s second law is the real law of motion.

Answer 1

Proof. When we can show that Newton’s first and third law are contained in the second law, then we can say that it is the real law of motion.

(i) First law is contained in second law : According to Newton’s second law of motion \(\vec{F}=m\,\vec{a}\)

Where, m = mass of the body on which an external force \(\vec{F}\) is applied and acceleration \(\vec{a}\) is produced in it.

When no external force is applied on the body,

i. e.,

When \(\vec{F}\) = \(\vec{0}\), then from equation (1), we get \(m\,\vec{a}\) = \(\vec{0}\), but as m ≠ 0.

∴ \(\vec{a}\) = \(\vec{0}\)

It means that there will be no acceleration in the body if no external force is applied. This represents that a body at rest will remain at rest and a body is uniform motion will continue to move along the same straight line in the absence of an external force. This corresponds to Newton’s first law of motion. So, first law of motion is contained in Second law of motion.

(ii) Third law is contained in second law : Consider an isolates system of two bodies A and B. Let the act and react internally.

Suppose

F_AB = force applied on body A by body B.

And F_BA = force applied on body B by body A.

When, \(\frac {d\vec{p}_A}{dt}\) = rate of change of momentum of body A.

and \(\frac {d\vec{p}_B}{dt}\) = rate of change of momentum of body B.

Then, from Newton’s second law of motion.

F_AB = \(\frac {d\vec{p}_A}{dt}\)

F_BA = \(\frac {d\vec{p}_B}{dt}\)

Equation (2) and (3) gives

F_AB + F_BA = \(\frac{d}{dt}(\vec{p}_A)+\frac{d}{dt}(\vec{p}_B)\)

\(\frac{d}{dt}(\vec{p}_A+\vec{p}_B)\)

Since no external force acts on the system (∵ It is isolated), therefore according to Newton’s second law of motion,

\(\frac{d}{dt}(\vec{p}_1+\vec{p}_2)=\vec{0}\)

or \(\vec{F}_{AB}+\vec{F}_{BA}=\vec{0}\)

or \(\vec{F}_{AB}=-\vec{F}_{BA}\)

Action = −Reaction

It means that action and reaction are equal and opposite. It is the statement of Newton’s third law of motion. Thus, 3^rd law is contained in the second law of motion.

As both first and third law are contained in Second law, so Second law is the real law of motion.