# Derive Newton’s first law from second law – prove

We already have discussed Newton’s first law and second law in other articles. These two laws deal with the motion of any particle in classical limit. First law gives the cause of motion and the second law gives the measurement of the force required to apply to make a particle in motion. That means these two laws are somehow interrelated. In this article, we are going to derive Newton’s first law of motion from Newton’s second law of motion.

If a particle of mass m has the acceleration a, then from Newton’s second law of motion we get, the force acting on the particle is F = ma.

Now, the acceleration of the particle is equal to the rate of change in velocity of the particle. Let, the initial velocity of the particle is u and after applying an external force F for the time t the velocity of the particle becomes v. Then the acceleration of the particle can be written as, $\small {\color{Blue} a=\frac{v-u}{t}}$.

Then the force on the particle is, $\small {\color{Blue} F=\frac{m(v-u)}{t}}$……………(1)

## What is the condition to establish Newton’s first law from the second law?

The statement of the first law of motion tells the condition itself. Newton’s first law of motion is applicable in the absence of externally applied forces on the particle. Under these circumstances, Newton’s first law is the same as the second law. We are going to prove exactly this.

## How Newton’s first law can be derived from Newton’s second law?

When there is no external force acting on the particle, F = 0.

Then from equation-(1) we get, $\small {\color{Blue} \frac{m(v-u)}{t} = 0}$

or, $\small {\color{Blue} \frac{(v-u)}{t} = 0}$

or, $\small {\color{Blue} (v-u) =0}$

Hence, $\small {\color{Blue} v=u}$……………..(2)

Thus, in absence of the applied force, the final velocity (v) of the particle is equal to its initial velocity (u). That means a moving particle always moves with the same velocity in absence of applied force.

Now, if the particle was initially at rest, the u=0. Then from equation-(2), its final velocity, v is also zero. Thus, in absence of applied force, the particle at rest will remain at rest forever.

Hence, Newton’s first law of motion is a consequence of Newton’s second law of motion.

## Summary

In this article, we became to know that Newton’s first law of motion is contained in Newton’s second law of motion. Then we found the condition to establish the first law from second law. At last, we went through the detailed derivation of Newton’s first law of motion from second law of motion.

This is all from this article on how to derive Newton’s first law from second law of motion. If you have any doubt on this topic you can ask me in the comment section.

Thank you!

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