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Newton's second law says that :

The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts. Mathematically it says,

$$\bbox[5px,border:1px solid black] { \frac{\Delta p}{\Delta t}\propto F } $$

Before I ask my question let me mention something:

One quantity A is said to proportional to another B, when the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio. Therefore, establishment of a proportionality relation beforehand requires the quantitative measurements of both A and B.

here comes my question i.e. in order to establish such a relation (∆p/∆t) ∝ F it requires quantitative measurements of both force and rate of change of momentum and as for the latter one can easily calculate the numbers but how come can someone calculate force ? Hence, how can one establish such a relation without having numbers for the force?

NOTE 1 : I'm considering the motion to be in right line(straight line) since for other cases one only requires to introduce vectors and the basic framework of the law remains the same.

NOTE 2 : For those who might think that how will I measure mass? It's quite simple. One can use a balance to measure the relative mass(as it is done in actual) but you might say hey,wait a minute we weigh stuff with a balance! so aren't we calculating the weight rather than mass and my answer would be NO!. Think about it once again. By using a balance you are making a relative measurement which is independent of the strength of the gravity so weather you are on moon, mars etc the balance weights will change by the same factor as the object you are measuring and so if you have some stuff in pan on one side of the balance and it requires(on earth) say 5 pieces of 1 kg(in other pan) to make pointer come to middle the same is true if u are on moon,mars etc.

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  • $\begingroup$ Do take in consideration that in order to measure the "relative mass" you still need to assume the force law $F=mg$, for there is no other reason why the ratio of the weights of two objects should be the same everywhere. $\endgroup$ – Othin Nov 6 '18 at 23:00
  • $\begingroup$ In all consistent systems of units, the proportionality constant is simply 1: $F=\frac{dp}{dt}$. $\endgroup$ – G. Smith Nov 7 '18 at 0:10
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If one were trying to prove Newton's second law, he would need an independent definition for the force (the second law cannot be regarded as a definition for it), with another formula. However, that's not the point. The second law tells you how an observer in an inertial frame is supposed to perceive the rate of change of momentum. By the first law, the velocity (and thus the momentum), will remain unchanged unless a force is acting on a body. Newton suggests (and that can't really be proven in the Newtonian framework) that the physical quantity that causes this change can be represented by a certain function (which he postulated to depend only on the coordinates which describe the spatial configuration of a system of particles).

When you create a mechanical model in the Newtonian framework, you need to suggest an expression for this function, a force law. This is the reason Newton couldn't derive his theory of gravitation from his laws of motion alone, he needed the guidance of Kepler's laws and the astronomical observations available at the time. That's also the reason the expression $\mathbf{F}=-k\mathbf{x}$ is called Hooke's law. Once such a law is suggested, every mechanical variable associated with the system, can, in principle, be calculated. As long as those results were not in contradiction with experiment, all was good.

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For example, you can measure forces using a spring scale.

To convince yourself that the spring scale really does measure force, you can do tests such as checking linearity.

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