Is Newton's first law included in the second law?
Yeah, you could say that.
Come on, one learns nothing from the 2 statements above. (Sorry I am curious to know too)
How are they related, and, if the 1st law includes the 2nd then what is the derivation?
Well, a statement of Newton's 2nd Law is that:
FNET = dp/dt
The 1st law can be thought of as a special case of the second law for a zero net force:
0 = dp/dt
==> p = const.
We can state this as, "In the absence of a net external applied force, an object in motion will remain in motion in a straight line at a constant velocity." (The Law of Inertia).
I've heard some versions of it say "...an object at rest will remain at rest and an object in motion will remain in motion in a straight line at a constant velocity." Although this is redundant (because "rest" is also a constant velocity...of zero), it is still nevertheless useful to state for clarity.
In spite of the above, I have seen people argue on this forum that Newton's 1st Law is more than just a special case of Newton's 2nd -- i.e. it is important to state it independently. I'm not sure, but I THINK that two reasons for this might be:
1. It was important for it to be stated, historically, when formulating mechanics, because at the time the notion that a force was NOT required to keep something moving was not taken for granted, nor was it obvious. This is because we tend to live in a world with friction, drag etc. and objects tend not to keep moving forever. Therefore, the fact that they ought to in an ideal case was neither obvious nor intuitive, and represented a great insight by Newton (as well as an important starting point for the investigation of physical laws).
2. It is not clear from the statement of Newton's 2nd Law whether a force is a necessary condition for a change in momentum, or merely a sufficient one. In other words Newton's 2nd doesn't make it clear that objects won't start moving either spontaneously or due to some other cause (other than an applied force). Therefore, it is important to to state the law of inertia explicitly in order to make this clear.
If anyone has comments on what I just said, they would be appreciated.
If an object does start moving from rest, it implies a change in momentum whose rate is given by the second law. Since the force is in the R.H.S of the second law, it can be inferred that any change in momentum should have a force responsible.
Okay, so you are saying that F = dp/dt should be correctly interpreted to mean, "an object's rate of change of momentum will be non-zero if and only if there is a non-zero net force being applied to it," (necessary and sufficient condition). I'll buy that, I'm just wondering what the correct interpretation is and why. I guess that, if you are right, that invalidates my second reason. So, what do others think about this issue? Also, what do others think about whether it is important that Newton's 1st Law exist independently of the 2nd (and why)?
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