On force being directly proportional to the rate of change of momentum

Question

Newton's second law says that $F_{Net}=\frac{dp}{dt}$.
I do realise that when the net force on a body is zero it keeps its current velocity, so one can infer that the force is somehow related to acceleration. But how do we move from this $F_{Net}=f\left(\frac{dv}{dt}\right)$ to simply the rate of change of momentum? Why would Newton think of momentum (i.e why include the mass constant)? Why not momentum cubed for instance?

EDIT: check Are Newton's "laws" of motion laws or definitions of force and mass?

Answer 1

Newton’s second law is the definition of force. This is what force is. If an object is accelerating, we say there’s a force acting on it. But in this form it’s utility is not evident. To see why this definition is useful (extremely) we need to have some physical phenomenon that causes acceleration. Luckily there’s a straightforward one, gravity.

At the time of Newton, Kepler had his laws derived empirically from the astronomical observations of Tycho Brahe. The genius of Newton was to figure out the underlying unifying structure in these laws: a differential equation between two masses that relates the acceleration (second order time derivative of displacement) to the masses and their relative displacement.

$$\frac{m_1m_2}{r^2}\propto m_{1}\frac{d^2r}{dt^2}$$

So the source of acceleration is gravity and it is governed by a second order differential equation. Nature has this underlying structure and that’s why this definition of a quantity called force is useful.

Answer 2

The force law can be seen more as a definition than law per se. What would be the force without the equation other than just vague description?

Anyway, the force should be some quantity that tells you how the interaction between bodies affects its movement. So how to construct good, useful definition of such quantity? From Galileos experiments Newton already had notion of first law of mechanics. That is, the interaction between bodies should affect not the velocity itself, but the change in velocity. Similarly, it seemed the physics is more or less the same everywhere so the force also should not depend on position of the body.

As first attempt, the force could be just an acceleration $F=a$. But this will not do. It is everyones experience that the more heavy the object is, the more force you need to apply, where "heavy" and "force" is now used in its day-to-day meaning.

The first attempt to include the objects properties would be to introduce some quantity $m$, that needs to be determined from object to object and define force as $$F=f(m,a),$$ where $f$ is a function to be determined.

The first thing I can do is concatenate bodies together and see how the function behaves. I make some device that will produce force. I don't know how big it is, because I did not define the force yet, but if the mechanism is the same, than it is reasonable to expect that exerted force should be also the same (for example I can use the same spring for every experiment). Quickly I find that if I use same mechanism on the object with mass $m_1$, then on the mass $m_2$ and then on concatenated object with mass $m$, then I will get: $$a_1/m_1=a_2/m_2=a/(m_1+m_2).$$ This holds for any kind of mechanism we use, so we can already see additivity of mass and that it holds $$F=f(ma).$$ Now, we can just write $F=ma$, because we can always redefine $F$ by taking $F\rightarrow f^{-1}(F)$. However, such definition might not be entirely useful. We would also like to know, that if we double our mechanism of force exerting (for example, by attaching the object to two identical springs instead of one), wheter the resulting acceleration will be also doubled. And we are in luck, because experiment shows it indeed will be.

Note: The last property is actually the seed of Newton's third law, so it is not so strange of a coincidence.

Answer 3

Isaac Newton used experiments (How did Newton discover his second law?) to figure out the relation between force $F$ and acceleration $a$. From the data he collected, he concluded that $F$ is proportional to $a$, a linear relationship. Any proportionality has a proportionality constant. Newton defined this proportionality constant to be inertial mass $m$. Therefore, $F = ma$.

Acceleration $a$ is the rate of change of velocity $v$: $a = \frac{dv}{dt}$. Therefore, we have $F = m\frac{dv}{dt}$. Using the linearity of the derivative, $F = \frac{d(mv)}{dt}$ or $F = \frac{dp}{dt}$.

So the answer to your question is that observations of the real world led to the linear relationship. This is how we figured out Coulomb's Law, etc. Physics is based on observations/experiments that then become analogous to mathematical axioms.