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The first law of newton tells us that a body shall remain unaccelerated when the net force acting on it is 0, but the second equation gives us the relation F=ma so, ain't the first law just an special case of the second law? So, what is the need for an separate law which is just an special case of the other one?

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There is a consideration that in Newton's time would not have made a difference, but nowadays it does.

What if space and time are not Euclidean?

What if we would live in a universe with circumstances such that you can be in inertial motion, but you are going around in a circle.

We can think of Newton's first law as expressing an assertion about the physical nature of space and time. The first law of motion then asserts:

  • Space is euclidean
  • Time flows uniformly
  • An object in inertial motion will move in a straight line, and in equal intervals of time it will cover equal intervals of distance.

That is: assuming that space is Euclidean, and that time flows at the same rate everywhere is in fact already a theory of physics.

Of course, back in Newton's time the scholars didn't think of it that way. Presumably it never occurred to any scholar of the time that space might not be Euclidean.

But in retrospect we can repurpose Newton's first law, thinking of it as an assertion about space and time.





In general, in my opinion there is plenty of reason to reformulate the laws of motion, so as to incorporate insights of centuries.

For instance, we can think of conservation of momentum in two contexts:

  1. The momentum of a single object that is in inertial motion is a constant.
  2. when two objects are exerting a force upon each other, causing change of momentum in the other object, then the total momentum is a conserved quantity.

It has become customary to think of 1. as being covered by Newton's first law, and to think of 2. as being covered by Newton's third law.


General consideration:

There are multiple ways to organize the fundamental concepts into a series of statements.

Exactly how that organization is implemented is not crucial. I'm guessing that's why there isn't a push to formulate a modernized set of laws of motion.

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The present point of view is that the first Newton's law establishes a special set of reference frames (the inertial reference frames) where the movement of a body non-interacting with other bodies is a rectilinear uniform motion. Becoming a statement on the reference frames is not a consequence of the second law.

Notice that historically, this was not the original interpretation. However, the evolution of the presentations of Newton's laws has a long history.

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When you apply the second law, 𝐹=𝑚𝑎 with zero force (𝐹 = 0), it shows that acceleration (𝑎) is also zero, which is exactly what the first law states: an object will not change its motion unless a force acts on it.

However, the first law is still important. It introduced the concept of inertia, the idea that objects resist changes in their motion. This helps us understand that motion doesn’t need a force to continue, only to change. Newton’s first law sets the stage for the second law by explaining what happens when no forces are acting on an object, emphasizing that force is needed to change motion, not to maintain it.

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  • $\begingroup$ but that also be understood from the same law that if the net force is 0, acceleration will also be 0 meaning it will not change it's state either or rest or constant motion $\endgroup$
    – Manish
    Commented Jun 15 at 14:20

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