Question

There are three Newton's laws of motion namely first,second and third laws.We can derive

• A. Second and third laws from the first law.
• B. First and second laws from the third law.
• C. Third and first laws from the second law.
• D. All the three laws are independent of each other.

Answer 1

The correct option is C Third and first laws from the second law.

Newton's second law of motion is $F=ma$

That is we have:-

$F=m(v-\frac{u}{t})$ thus, $Ft=mv-mu.$

Now,When $F=0$, then $v=u.$ That is the absence of force,The object continue to move with same velocity

throughout.

Now,when $F=0$ and $u=0,$ then $v=0.$ That is , an object at rest if no force is acting on it.

Thus,

Newton's first law is derived from the second law.

Now let us consider a system of $2$ bodies $1\&2$ and considers that there is no external force acting.

Now let $F_{12}$ be the force acting on $2$ by $1$ and $F_{21}$ be the force acting on $1$ by $2.$

The rate of change of momentum of $1=\frac{d{p}_1}{dt}$ and rate of change of momentum of $2=\frac{d{p}_2}{dt}$

Thus, according to Newton second law of motion $F_{12}=\frac{d{p}_2}{dt}$ and $F_{21}=\frac{d{p}_1}{dt}$

Adding both the above equation, we get:-

$F_{12}+F_{21}=\frac{d{p}_2}{dt}+\frac{d{p}_1}{dt}=d(p_2+p_1)/dt$

We know that, no force is applied.

Thus momentum change will also be $0$ change in velocity occurs.

Thus $\frac{d(p_1+p_2)}{dt}=0$

Therefore, $F_{12}+F_{21}=0$

That is $F_{12}=-F_{21}$

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

Hence,

option $C$ is correct answer.