The nature of the physical law

Remember school, we had to memorize all those physical laws and "spew" them back when asked... and of course we forgot them almost instantly. We were not thought how to think, but how to memorize. One more reason for me to try to explain why "laws" were discovered/invented the way they are. In this article I will use the famous Newton second law from classical physics as example, you will see it is very easy to come up with very simple reasoning to prove it, dont be afraid ;). And there is almost no formulas only at the end to make thing clearer, but they are not necessary for understanding the principle. I will try to touch on areas normally missed in teaching those laws, which I think is more important than mindless memorizing. I think the reasoning behind is much more interesting than the bare mathematical formula, because the math formula is there to allow us compact representation of the whole "enchilada" that goes behind the scenes, not to cause us pre exam nightmares.

BTW, "law" is a wrong word to use, because first a "law" is a man made "construct" and second because with physical "laws" there is never 100% certainty of them being the absolute TRUE. The "physical laws" are as true as our understanding of nature goes. Only a belief can declare something to be 100% TRUE. In Science we use Axioms, which is supposed to be something so obvious as to not to require any proof, the difference being as I said already that Axioms are true up to point of our current understanding of nature.
We use the term "velocity" instead of "speed". Using it is more correct thing to do in some circumstances. Velocity is simply speed with direction - a vector, speed is scalar f.e. negative velocity means that the object just sped up in the opposite direction. If somebody ask you how fast did you drive your car, you normally say something like this :
 - I drove with 60 km/h.
This is speed, but if you say :
 - I drove with 60 km/h in North direction !
OR :
 - I drove with 60 km/h towards city X !
you are describing velocity. See, quantity and direction.

OK, let's start....
Today's quest is to find some connection between action upon an object and what happens with it and to find the quantitative relation of this connection.
Imagine rolling a ball on the ground !
The ball rolls let say 1 meter and stop. What we will do is clear some dirt and roll the ball again with the same strength, the ball will go a little further, say 1.5 meters. Now what if we roll on a glass or some other polished surface again with the same strength .... the ball will go even further. If we extrapolate our thought experiment and use surfaces with even less friction the ball will roll further more.... what if we continue to do so indefinitely i.e. we roll on frictionless surface ... the ball will roll forever, with the velocity it reached after we released it !!
(There is nothing to slow down the ball, and the velocity is constant. To get higher velocity we have to roll harder, more on this later). You may say to me that frictionless surface does not exists, to which I won't object, you thought you got me ;).
We do not try to prove that such surface exist ... we simply abstract our thought experiment to such a point, so that we can find a general pattern.
Once we get a pattern like this, we write it down, then when we have real world problem we use the abstraction and add the complications we know exist in reality afterwards to get satisfactory result. If we tried to figure all obstacles and calculate all the outside influences beforehand we won't live long enough to see the result, just for one single observation or experiment.
OK. With that out of the way, let's think of some other experiment which can give us a clue is the pattern we just found applicable to other physical phenomena. What if we throw a ball instead of rolling it ?
What would happen is that it will fly some distance and drop on the ground. This normally happens with all objects we throw or just drop ...i.e. they end on the ground some distance away. We reason that there is something that attracts the objects to the ground. Us throwing the ball is not the reason it drops, because if we just drop the object it will still end on the ground. (Remember we want to find the connection between us throwing or rolling and what happens with the ball if we remove any outside influence, even things such as gravity).
Let's assume that this attraction disappears somehow, what will happen then, the ball will fly further. It will slow after awhile, because of our old friend the "friction", this time air friction. We repeat the same reasoning and lower the friction, less and less... and again in some infinite ideal scenario the ball will fly forever with the velocity it reached after we released the ball.
If I have to give again example of why this abstraction is useful. Let's say this abstraction leads me to a rule of a way to calculate the distance to which a cannon ball will fly. What I do again I use this calculation, make similar reasoning for the attraction of object to Earth, then another one for the friction of air ... and after I do alot of tests, I will probably add or multiply or divide those different rules, depending what is the relation between them. Remember at the end of the day whatever basic physical law we come up, it has to be proved by experimental data up to some point of precision.
Time again for another definition... what we did was to roll and throw a ball. We can also hit it, push it, pull it with something. All those actions in physics are called FORCE (from Greek : strength).
We are ready to give definition of our first conclusion :
If we apply a force to an object it will reach a specific velocity and move forever with that velocity, unless there is outside forces acting upon the same object.
That is our first Axiom ;), see even that it is an axiom, we still tried to reason about it.
Definition : Change in velocity is called "ACCELERATION".
So from our axiom it follows that applying a force changes velocity. This was our crux point ... I will repeat again, applying a force on a object CHANGES its velocity.
In our previous experiments we had one of our variables constant i.e. we threw OR rolled the ball with the same strength i.e. we applied the same force every time.
What will happen if we throw harder in our frictionless environment, the ball will speed up to higher velocity. If we throw weakly the end velocity will be lower.
In cases like this we say that one measure is correlated to the other i.e. in our case acceleration is correlated to force. To state this we use the tilde sign '~', like this :
This does not mean equality it just means that if one of the measures changes the corresponding one also changes, but not necessary in the same scale, just in the same quantity-direction. F.e. let say that if I double the force that I apply to the object and the acceleration changes by half much. If that is the case I can write :
 `1/2 * ACCELERATION = FORCE`     (not yet)
We can not do this yet, because the generated acceleration is not dependent only on the strength of the applied force, but also on the mass of the object. You see I used the word 'mass', instead of 'weight'.
Side note : In physics there is this subtlety again, we use the word "mass" when we ascribe a heaviness of an object. Why should we do that ? The reason is that the weight is product of gravitation i.e. one object will have different weight on Mars and different weight on the Moon and different weight on Earth, but its mass will always be the same (It is even subtler than this, but the current explanation will suffice for now).
Weight is a force and mass is this "constant" attribute describing the heaviness/massiveness of a body. In everyday life we do not have to do this distinction because normally we don't care how much somebody weight on Mars.
Our experiment again, if we throw heavier objects with the same force, the acceleration produced will be smaller and vice versa. This means that acceleration is reciprocal to the mass and we write it like this :
The bigger the mass, less acceleration we get and vs. versa. Now because the ACCELERATION (change in velocity) produced is dependent on both the applied FORCE and the MASS of the object, we can summarize our observations like this :
Because in most of the cases we are interested of what force was acted upon an object i.e. we know the mass and acceleration and we are looking for the force, we can rewrite this correlation rule like this :
Remember the trick to make reciprocity into equality, I mentioned earlier ? We can use a constant to do that, but we wont... ;), I was just showing you one way of doing it. Instead we will use another dirty trick practiced by the physicists, namely we will use 'native' units.
Here is how it works. First let see how we can measure the values on the right hand side of the equation.
For MASS we can use : kg, g, lbs, oz ....
  ACCELERATION = VELOCITY / TIME  (remember: change in velocity)
  VELOCITY     = DISTANCE / TIME  (distance traveled for a time)
    i.e. Distance : km, m, cm, mile, inch, foot
    and Time : sec, hour, min
    => Velocity : km/h, m/s, mile/h
  which give us for Acceleration : m/s^2, km/h^2 ..
    (the first one pronounced meter per second per second OR meter per second squared)
OK, now that we have those means of measuring the trick is to use correct units, based on our formula. Let say kg with meter and second, so if we use those we can say :
`kg*m/s^2 = MASS * ACCELERATION`
which if you look is exactly right. What does it mean is that as long as we measure the strength of a force with the unit of measurement : kilogram x meter per second per second, we can write :
OR as it more commonly known : >> `F = m*a` <<
Did you saw what we did ? We used the correct units to convert correlation to equality, that is the magic trick. If there is big discrepancy in the scale of the different sides of the equation we can use/invent constants to get better equality. Most of the time such constants are invented based on experimental data OR logical deduction. And again the purpose of equations is to capture a pattern.
Btw not that you will be surprised by it, but the unit of measurement `kg*m/s^2` other name is Newtons. So we measure force in Newtons.
Other way to understand this measure is that Force of 1 Newton is required to accelerate an object with mass 1 kg at a rate of 1 meter per second per second.
We don't have to use exactly this measure, if we measure smaller forces we can use f.e. `gram * cm` / `sec^2`, this one is not called Newton, but "dyne".
So here we have it with simple reasoning : >> `F = m*a` <<
Remember the formula is just so that we can easily do specific calculation, what was our guiding principle was to reason some basic axiom, with good deduction to capture a pattern.