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

Hint Newton's first law states that the object at rest or in other words it is in motion will remain unless the object relates a net external force But the newton's Second Law of Motion relates force, mass, and acceleration.

Complete step by step solution
Newton’s second law of motions states that the net force applied on a body is equal to the rate of change in its Momentum.
We know that, $F = ma$
$a = v - \dfrac{u}{t}$
Substitute, $F = m\left( {v - \dfrac{u}{t}} \right)$
Simplify, $Ft = mv - mu.$
Now, when $F = 0$then $v = u.$ This shows the absence of the force. So, the object continues with the same velocity.
Now, when $F = 0$ and $u = 0$,then $v = 0.$That is, an object at rest if no force is acting on it.
Therefore, first law is derived from the second law.
Now, let us consider a system of two bodies, one and two bodies, there is no force acting on it.
Now ${F_{12}}$ be the force acting on 2 by 1 and ${F_{21}}$ be the force acting on 1 by 2.
The rate of change of the momentum of $1 = \dfrac{{d{p_1}}}{{dt}}$ and the rate of change of the momentum of $2 = \dfrac{{d{p_2}}}{{dt}}$.
Adding both, we get
$\dfrac{{d\left( {{p_2} + {p_1}} \right)}}{{dt}}$
Therefore, the momentum change will also be 0 change in velocity occurs.
Therefore, ${F_{12}} + {F_{21}} = 0.$
Thus, Newton’s third law is proved with Newton’s second law.
Therefore, Newton’s first and second laws with second law.

Hence, from the above options, option C is correct.

Note Newton’s first law states that the body states at rest, if it is at rest and moves with a constant velocity if already moving, until a net force is applied to it. In other words, the state of motion of the body changes only on application of a net non-Zero force.