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I came across this translation of Newton's Second Law:

"The alteration of motion is ever proportional to the motive force impressed and is made in the direction of the right line in which the force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion..... "

I was wondering if there is an experimental proof of Newton's Second and Third law. I believe that the first law is more of an empirical fact.

But I don't get how is it possible to make quantitative claims about forces (and motion) when (to my knowledge) the only way to quatify a force is through the Second law. Newton's Third law ( I believe) follows from the second law and the Law of conservation of momentum.

Or is Newton's Second law the quantitative definition of force? Could somebody please clarify?

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    $\begingroup$ It's not clear to me either what "double the motion" would mean in this context. $\endgroup$
    – JMac
    Commented Jun 4, 2019 at 15:59
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    $\begingroup$ This looks like a duplicate of physics.stackexchange.com/q/292309 $\endgroup$
    – TimWescott
    Commented Jun 4, 2019 at 16:06
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    $\begingroup$ Please read this answer of mine physics.stackexchange.com/q/439193 .Laws are like axioms for physics theories. With the same logic, where in mathematics axioms can become theorems and theorems axioms. $\endgroup$
    – anna v
    Commented Jun 4, 2019 at 16:25

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We do know that Newtons second law is "right" experimentally in the world we normally experience with our human senses. If things are very fast need the speed of light, then Einsteins theory of special relativity is better. If dealing with atoms, Quantum mechanical may be more correct. If near a black hole then General relativity. But for our normal day to day engineering and physics we know Newton's laws work quite well through observation and experiment and we know it is useful over many orders of magnitude from very small objects, in some cases even atoms to the motion of planets and galaxies. Almost all engineering designing of structures, or moving stuff uses Newtons laws in some form in its design.

The simplest experiments to show Newtons second law are experiments like a pendulum, a mass on a spring, or the collision balls on a pool table and assuming there is little friction. If you change the masses or forces and plot distances, velocities and accelerations you see that Newtons second law works.

At a more fundamental level, if you accept that energy is conserved and that momentum is conserved, then you find that force is the derivative of momentum, and in that way you can also derive F=ma, (if a rocket also keep track of the changing mass).

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    $\begingroup$ I would like to add that even if we take general relativity or quantum mechanics, it can be shown that at the classical limit (not so fast, not so small, not so heavy) we recover Newton's laws. $\endgroup$
    – Mauricio
    Commented Jul 21, 2022 at 12:05
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    $\begingroup$ That’s an important point! $\endgroup$
    – UVphoton
    Commented Jul 21, 2022 at 12:20
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Newton's laws are more akin to thought experiments. However, the sandwich on my plate remained still for 30 minutes until I picked it up. So this could be counted as experimental evidence.

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Newton basically was making definitions.

A force is anything that changes the velocity of something you measure. With no forces it will continue in the same direction at the same speed. If the direction or speed changes then by definition a force is changing it. By definition, if it changes twice as fast then there is twice the force changing it.

By definition, there is always an equal and opposite reaction. Any time we do not observe an equal and opposite reaction, there must be something invisible that is carrying away the reaction that is not observed.

So for example when some nuclear reactions that produced beta particles were observed to produce beta particles with many different velocities, not at all the same energy for each, not the same momentum, it was hypothesized that an undetectable particle -- a neutrino -- was being produced which carries away the rest of the energy and momentum. Similarly when other reactions use more energy than is there, undetectable particles must be providing that energy.

And it turns out that in places where the undetectable particles are produced, more undetectable particles are being consumed also. This is considered to be a way to detect the undetectable particles.

Since Newton's laws are true by definition, it is not possible to find examples where they are not true.

However, Maxwell's equations described electromagnetic force, and it behaves strangely. The exact same force that has a big effect on things that are stationary or moving slowly, has a smaller effect on things that are moving fast. Twice the force does not have twice the effect. It's because electromagnetic force itself travels at lightspeed, and can't make anything travel faster than lightspeed. Newton's laws need a correction for that. For awhile people tried to make the correction be that things which travel fast have increased relativisitic mass, and then Newton's laws could work unchanged. But that caused other problems so they mostly don't talk that way now.

Also, electric force travels at lightspeed, and it depends on the velocity of the source when it leaves the source, and the velocity of the target when it arrives at the target. That can play hell with the equal and opposite reaction. Two charges are at rest and force leaves each of them. Then they scurry around in different directions, and come to rest again before the force reaches them. The force arrives from different directions at different distances and different times. Is that going to be equal reaction? No.

But Newton's Laws still work perfectly for forces that are independent of electromagnetism and which have instant effects.

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