[Physics] Definition of force, kinetic energy and momentum

Question

I've edited the post. Q1 and Q4 are the important ones but I didn't delete Q2 and Q3 since some older answers would not make sense anymore.

To begin with, the formula of the kinetic energy $T$ is $\frac{mv^2}{2}$. Furthermore momentum is conserved $\Sigma m_{i}\vec{v_{i}}=const.$ Then you have the definition that force is the change of momentum with respect to time $\vec{F}=\dfrac{d(m\vec{v})}{dt}$. I've read the chapters concerning mechanics of Physics for scientists and engineers by Giancoli and the Feynman Lectures. Giancoli introduces the arbitrarily work as $W=\int\vec{F} \cdot d \vec{s}$. From this definition of work he derives the kinetic energy to be $\frac{mv^2}{2}$. In contrast to that, in the Feynman lectures you never get a derivation of $\frac{mv^2}{2}$, but it is shown that $\dfrac{dT}{dt}=\vec{F}\cdot\vec{v} = \vec{F}\cdot\dfrac{d\vec{s}}{dt}$. Then it is shown that $dT=\vec{F}\cdot d \vec{s}$ and as a consequence $\Delta T = \vec{F} \cdot \vec{s}$ which is called work. http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S1

Now I've got some questions:

Q1 Is $F=\dfrac{d(mv)}{dt}$ just an arbitrary definition or is there something "more" behind the formula for force?

Q2 Is $W=\int\vec{F} \cdot d \vec{s}$ just a definition or is there something more behind? I mean, can you derive the formula for work not by taking the formula for the kinetic energy as given.

Q3 How to derive the formula for the kinetic energy and work only form the conservation of momentum $\Sigma m_{i}\vec{v_{i}}=const.$?

Q4 How are work and kinetic energy defined? I have found both: A) Kinetic energy is doable work and work is $F=\dfrac{d(mv)}{dt}$. B) The formula for kinetic energy is: $\frac{mv^2}{2}$. Then, after some maths it follows that $\Delta T = \int \vec{F} \cdot d\vec{s}$. It doesn't make sense to define 2 things in that way. This would be circular logic.

P.S.: I am not a native English Speaker so feel free to edit.

Answer 1

Feynman makes a point of stating explicitly, in vol. 1 of his Lectures on Physics, that $F = \frac{d(mv)}{dt}$ is not the definition of force. In section 12-1 he states

If we have discovered a fundamental law, which asserts that the force is equal to the mass times the acceleration, and then define the force to be the mass times the acceleration, we have found out nothing.

A bit later he states

The real content of Newton's laws is this: that the force is supposed to have some independent properties, in addition to the law $F = ma$; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law $F = ma$ is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple.

I found the following comments from Terence Tao, on the topic of how physics models work, to be enlightening:

Terence Tao - @Pietro: the way mathematical or physical models work, one assumes the existence of a variety of mathematical quantities (e.g. forces, masses, and accelerations associated to each physical object) that obey a number of mathematical equations (such as F=ma), and one also assumes that the result of various physical measurements can be computed in terms of these quantities. For instance, two physical objects A_1, A_2 will be in the same location if and only if their displacements x_1, x_2 are equal.

Initially, the numerical quantities in these models (such as F, m, a) are unknown. However, because of their relationships to each other and tophysical observables, one can in many cases derive their values from physical measurement, followed by mathematical computation. Using rulers, one can compute displacements; using clocks, one can compute times; using displacements and times, one can compute velocities and accelerations; by measuring the amount of acceleration caused by the application of a standard amount of force, one can compute masses; and so forth. Note that in many cases one needs to use the equations of the model (such as F=ma) to derive these mathematical quantities. (The use of such equations to compute these quantities however does not necessarily render such equations tautological. If, for instance, one defines a Newton to be the amount of force required to accelerate one kilogram by one meter per second squared, it is a non-tautological fact that the same Newton of force will also accelerate a two-kilogram mass by only one half of a meter per second squared.)

If one has found a standard procedure to compute one of these quantities via a physical measurement, then one can, if one wishes, take this to be the definition of that quantity, but there are multiple definitions available for any given quantity, and which one one chooses is a matter of convention. (For instance, the definition of a metre has changed over time, to make it less susceptible to artefacts.)

In some cases, it is not possible to measure a parameter in the model through physical observation, in which case the parameter is called "unphysical". For instance, in classical mechanics the potential energy of a system is only determined up to an unspecified constant, and is thus unphysical; only the difference in potential energies between two different states of the system is physical. However, unphysical quantities are still useful mathematical conveniences to have in a model, as they can assist in deriving conclusions about other, more physical, parameters in the model. As such, it is not necessary that every quantity in a model come with a physical definition in order for the model to have useful physical predictive power.