I have read this great answer (first one) by joshphysics: Are Newton's "laws" of motion laws or definitions of force and mass?

It cleared all my confusion regarding what mass actually is and how we define force. What is missing though is how has Newton reached these three observations? For example, what experiment could he make to discover that the ratio of accelerations of two objects in an inertial frame of reference is always constant? You would need to somehow measure the acceleration of both objects in the very very short time they bump into each other, how can it be done?

  • $\begingroup$ "accelerations ratio of two objects in an inertial frame of reference is always constant?" What do you mean by acceleration ratios? And why do we have to be looking at colliding objects? $\endgroup$ – Aaron Stevens Aug 14 '18 at 17:51
  • $\begingroup$ @AaronStevens Third Law. "If two objects, sufficiently isolated from interactions with other objects, are observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant." how could you derive this fact from experiments? $\endgroup$ – Venuce Aug 14 '18 at 18:12
  • $\begingroup$ Physical laws are never derived from experiment. $\endgroup$ – lalala Aug 16 '18 at 16:49
  • $\begingroup$ @lalala Is the key word in your challenging statement "derived", or have I misunderstood? $\endgroup$ – Philip Wood Aug 16 '18 at 21:53
  • $\begingroup$ @PhilipWood yes. Although I dont consider the statement challenging. Instead of "derived" I think "suggested", "inspired" or "motivated" might provide a better understanding how physics is really created (although teaching books seem to imply you can derive stuff from experiment....) $\endgroup$ – lalala Aug 17 '18 at 9:06

You don't need to measure the acceleration during the collision. For one thing, the force between the bodies (that is $F_{BA}$, the force that A exerts on B, and $F_{AB}$, the force that B exerts on A) will continuously change in magnitude during the collision, and so will the bodies' accelerations. But $F_{BA}=-F_{AB}$ at all times (N's Law 3) so, according to N's Law 2, $$\frac{dp_B}{dt}=-\frac{dp_A}{dt}.$$in which $p_A$ and $p_B$ are the momenta of A and B. This equation integrates up to give The Law of Conservation of Momentum applied to two bodies, namely$$p_{A}+p_{B}= \text{constant}.$$If the bodies start from rest and spring apart (a so-called 'explosive collision'), then we have$$p_{A}=-p_{B}\ \ \ \ \ \text{that is}\ \ \ \ \ m_{A}v_A=-m_{B}v_B$$in which $v_A$ and $v_B$ are the bodies' velocities as measured any time after the collision, even when the bodies are well separated – provided friction hasn't slowed them down.

So you can test the joint consequence of N's 2nd and 3rd laws for bodies of fixed mass, by showing that the ratio of their velocities after an explosive collision is constant. I don't think Newton could have done this with any precision, as he didn't have access to air-tracks and suchlike, so friction would have spoiled his experiments. [That's yet another mark of Newton's genius: he was able to figure out the laws of dynamics from how planets and satellites moved, and to realise that the same laws applied here on Earth – though he couldn't test them here anything like adequately.]

If you want to test N's 2nd law by itself you might pull a trolley by a single spring stretched to a known extension, then by two such springs stretched to the same extension, in parallel with each other and so on, measuring the acceleration each time. Friction compensation needed. Although it's possibly not beyond dispute that two identical, identically stretched springs in parallel will exert twice the force that a single one exerts, I find it very hard to doubt, as it's so closely bound up with my concept of force (a concept not wholly captured by "mass $\times$ acceleration" or "rate of change of momentum"). [Note that I'm not relying on Hooke's law.]

[The springs and trolley experiment was (and possibly still is) done in many schools in the UK to teach pupils in their mid teens about forces and Newton's laws. I believe it originated with the Nuffield science teaching initiative.]


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