http://www.straightdope.com/mailbag/mequalopposite.html
Actually, that sounds like a trick question to me – The equal and opposite force is required by the third law, not the second. Newton’s second law is generally presented as “the sum of the external forces acting on an object is equal to its mass times its acceleration,” or just “F = ma” for short.
What you quoted is a fairly silly answer anyway, because it’s not an equal and opposite force; it’s an equal an opposite reaction. Which is basically conservation of momentum.
A bit of trivia. What Newton actually wrote for the third law is more accurately translated into modern symbolism as F=dp/dt. F=ma was just assumed by later physicists. It’s interesting, because F=dp/dt doesn’t break modern relativity.
Nitpick - that’s the second law. And FTR, physicists still do use F=dp/dt in many situations.
Even if it said third law, I would still want to smack my physics teacher upside the head if he asked it. Am I supposed to be looking for the equal-and-opposite force to the force mentioned, or am I supposed to be looking for the equal-and-opposite force that is keeping the cup from moving? Because even after I looked at the answer I was supposed to give, it took me half an hour to think of why my answer (the normal force) is not correct. I suppose since the question didn’t say anything about the movement or resting of the cup, I’m supposed to answer with the one given with the article. But it’s still not fair.
I’m not even being graded on this, and it’s still pissing me off.
Actually, any physicist worth his salt doesn’t assume F=ma. He shows that it is reasonable to use a model in which the given body has a constant mass. Then
F = dp/dt = d(mv)/dt = dm/dtv + mdv/dt = 0+ma = ma
Only if you use four-momentum as opposed to simple three-momentum. Mainly because dm/dt =/= 0 relativistically.
Incidentally, it’s worth noting here that dm/dt is nonzero not because of the dm part, but because of the dt part. Time, like space, is subject to relativistic distortion. If you want to use a more reliable notion instead of “time”, you can use “proper time” or “proper interval”, symbolized by tau, which is not distorted. And dm/d(tau) = 0.
Getting back on topic, the Third Law always, under all circumstances, requires a pair of equal forces (despite the typo, this is what the Staff Report is about). This is even true in the context of relativity or quantum mechanics, and is true in all reference frames and bases. The second law, however, requires equal and opposite forces if and only if you have no acceleration, which is only a special case of the laws of motion. If you have acceleration, then the Second Law will absolutely not give you a pair of equal and opposite forces.
Finally, I must say that I don’t think that the Staff Report will be helpful to the questioner. When the question includes statements like “If I am going forward with the force of 10 newtons…”, I think that exposes some pretty fundamental misunderstandings of just what force, say, is. I’m not sure whether it’s possible to correct misunderstandings that deep in the course of a single Staff Report, or even a message board thread.
I like these types of threads even though I don’t have the education to add to them ( most of the time it takes me about a week just to read them.)
I can add that I really didn’t understand the question… specifically the part about the newtons.
I figured the question was just beyond my understanding of the subject-- and may yet be-- but now I have to ask, does the question even make sense the way it’s stated?
… dm/dt is nonzero not because of the dm part, but because of the dt part…
ow! ow!
curse you, Chronos. You broke my brain!
I wonder though: if an object could push against the entire material universe, wouldn’t it appear to be moving with no balancing reaction? Relativity says that there is no absolute inertial framework, so you can only measure an object’s motion against other objects. So if you’re pushing against everything else, up to and including the microwave background, wouldn’t that look like motion coming out of nowhere? Or do you have to drag in General Relativity at that point and the whole question of where inertia comes from?
You already have dragged in General Relativity, I’m afraid.
No, it doesn’t. You can’t be “going forward with a force of 10 newtons.” Force must describe the interaction of two bodies. For instance, you could push on a wall with a force of 10 newtons. It seems to me that the questioner may be confusing force with momentum.
Uuummm … like any good c++ programmer, I renumbered Newton’s Law’s starting with 0.
Ah, of course! 
The force pushing against the entire universe would still propagate at a finite speed (c, at best), so there would be some sort of ‘ripple’ showing where the action-reaction was. I suppose that would look like the GR/source of inertia thing you mentioned, but I don’t think you can avoid it.
I can’t possibly justify what I just said, but I still think it’s essentially correct.
It sounds like what you’re saying is you’d want the force exerted by this object to accelerate every other object in the universe by the same amount. For this to be the case, it would have to simultaneously push with a different amount of force on each object in the universe, depending on their masses. So what you’d need is some sort of force that depended on mass but not on distance, like if you could take away the r^-2 dependence of gravity. As far as I know no force like that exists, but if one did exist then I guess its effect would look like something accelerating without pushing on anything. I guess.
I’m not sure why bringing in the whole Universe makes things any more interesting. If I’m pushing on the Universe with 5 Newtons, then the Universe is pushing on me with 5 Newtons. Admittedly, that 5 Newtons will have a much greater effect on me than it will on the rest of the Universe, but this is qualitatively no different from when I push on the Earth and the Earth barely reacts.
And Karen, my compliments. That was one of the all-time greatest weaslings I’ve seen :D.
I think the point was not that the motion of the universe will be negligible because it is so big, but that the motion of the universe couldn’t be observed because it wouldn’t be moving relative to anything (except the person doing the pushing, obviously). Although, again, this would only be true if you accelerated everything by the same amount – just pushing on everything with a force of 5 newtons wouldn’t do the trick. At any rate, I don’t think it’s possible to do, so if the point was to find an exception to Newton’s third law, I don’t think this qualifies.