Question

State Newton’s second law of motion and shown that the first law of motion can be mathematically started from Newton’s second law.

Answer 1

Hint: Newton’s built Galileo's ideas and laid the foundation of mechanics in terms of three laws of motion that go by his name. Galileo’s law of inertia was Newton’s starting point which formulated as the first law of motion.

Complete answer:
Before going to Newton’s second law of motion it is important to understand some terms i.e. momentum. It is product of its mass $m$ and velocity $\overrightarrow{v}$ and it is denoted by $\overrightarrow{p}$,
$\overrightarrow{p}=m\overrightarrow{v}$
Newton’s second law relates with momentum.
Newton’s second law can be stated as,
The rate of change of momentum of a body is equal to the applied force and takes place in the direction in which the force acts. The average force acting on a body during a momentum change is
$\overrightarrow{F}=\dfrac{\Delta \overrightarrow{p}}{\Delta t}$ --(1)
Taking limit$\Delta t\to 0$, the term $\dfrac{\Delta \overrightarrow{p}}{\Delta t}$ becomes $\dfrac{d\overrightarrow{p}}{dt}$
So, $\overrightarrow{F}=\dfrac{d\overrightarrow{p}}{dt}$ --(2)
For a fixed mass $m$,
$\dfrac{d\overrightarrow{p}}{dt}=m\dfrac{d\overrightarrow{v}}{dt}$ --(3)
where $\overrightarrow{v}$ is the velocity of the body.
From (2) and (3), we get,
$\overrightarrow{F}=m\dfrac{d\overrightarrow{v}}{dt}=m\overrightarrow{a}$ $\left( \because \dfrac{d\overrightarrow{v}}{dt}=a \right)$
where $\overrightarrow{a}$ is the acceleration of the body.

Let us now consider Newton’s first law. If the net external force on a body is zero, its acceleration is zero i.e. acceleration can be non-zero only if there is a net external force on the body.
This law can be explained or mathematically started using Newton’s second law i.e. equation(2)
$\overrightarrow{F}=m\overrightarrow{a}$ -- (Newton’s second law)
If $\overrightarrow{F}=0$ implies $\overrightarrow{a}=0$ which is consistent with the first law of motion.
Hence, from the above equations and laws it is clear that Newton’s first law of motion stated mathematically by Newton’s second law.

Note:
Students must get into the habit of writing Newton's law of motion in the vector form as the vector form gives a better idea of the various nuances regarding the directions and components of the physical quantities involved. For example, in Newton’s second law $\overrightarrow{F}=m\overrightarrow{a}$ implies that the acceleration of the body is in the direction of the force applied.