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My textbook mentions that one of the properties of an inertial frame of reference is that Newton's first law must be valid in it. I'm a bit curious because it only mentions Newton's first law and say, not its second or third so I'm just wondering if there's any situation in an inertial frame of reference where the latter two laws don't apply?

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In an inertial frame of reference, it is necessary that all laws of Physics hold; in fact, this is one of the basis that forms Special Relativity. Hence, for an inertial frame of reference, the other two laws hold. The reason your book only mentions Newton's First Law is that we use this as a definition for an inertial frame; it's like we're just saying an inertial frame is one that travels at a constant velocity (which is a simplification, but that is the general idea).

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