Derive Newton's first law from second law?

According to Newton’s second law of motion, the rate of change of momentum of a body is directly proportional to the applied force and it takes place in the direction in which the force acts.

Consider a body of mass m moving with velocity v, then the linear momentum of the body is given by

p = mv

Now from, Newton’s second law,

F ∝ \(\frac{dp}{dt}\)

⇒ F = k \(\frac{dp}{dt}\)

Where, k is constant of proportionality.

As, p = mv

⇒ F = k \(\frac{d(mv)}{dt}\) = km \(\frac{dv}{dt}\)

But, \(\frac{dv}{dt}\) = a, acceleration of the body

The value of the constant of proportionality k is considered as 1 for simplicity.

By taking k = 1, we get

F = ma

According to Newton’s first law of motion, everybody continues in its state of rest or uniform motion in a particular direction until and unless an external force is applied to change that state.

Now, if net force, F = 0, then by Newton’s second law acceleration, a = 0.

This shows that, if there is no force acting on the body, its acceleration is zero i.e. the body will be in its state of rest or uniform motion. Hence, Newton’s first law is derived from the second law.

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