Is velocity real?

There is no conservation of velocity because that's just the way this universe works incidentally, we noticed that momentum is conserved in isolated systems and formed a law.

Whether or not you can get an intuition for something is purely subjective, your intuition for an entity does not determine its 'existence'.

Frankly I find it hard to describe reality, in everyday terms we just use intuition. I feel this is just a philosophical question not a physical one.

What is 'Real'? How do you define 'Reality'?

This is not what we deal with in Physics... We just try to get an intuition with the physical quantities while dealing with them. So the question is about whether you can get an intuition for velocity or not.

Conservation of an entity is purely incidental and it's just a quality of the universe we live in.

Displacement which you understand, and time which you understand, form the ratio velocity which is displacement divided by time.

Consider an object that travels $100 m$ in ten seconds. That means its moved from its starting point to another point $100 m$ away in $10 s$. We then say that it’s velocity is

$$v = frac{100m}{10s} = 10 m/s$$

meaning it travels $10 m$ in one second. If we then say it travels $1000 m$ in say $50 s$ then its velocity is

$$v = frac{1000m}{50s} = 20 m/s$$

meaning it travels $20 m$ in one second.

but no law of conservation of force or velocity. Why not?

In physics, conserved quantities are those which do not change over time and can be determined from symmetry operations. If we call an arbitrary dynamical quantity $K$ and if $K$ does not change over time, then

$$frac{dK}{dt} = 0$$

and $K$ is a conserved quantity and is associated with a symmetry in the physics of the system. After what we noted above, the same cannot be true for a quantity like $v$ which is $dx/dt$ (and also force using the same explanation and force is $propto dv/dt$).

For example, the fact that energy does not change over time, and is therefore conserved, is a result that can be obtained by time translational symmetry of the system.

Another example is the fact that momentum is conserved, and is a result that’s obtained by translational symmetry of the system. And angular momentum conservation comes from rotational symmetry of the system.

There is no such operation for quantities like force or velocity (indeed it makes no sense to even ask if this is true for such quantities, since they are not individual properties of particles/systems).

It would also help you if you had a read of this article regarding Noether’s theorem which discusses symmetries and conserved quantities. The important points:

“Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law....The action of a physical system is the integral over time of a Lagrangian function (which may be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.”

When we do intuitive physics we use the categories our brains come "prepackaged" with, things like size, duration, force, warmness or colour. As we refine our understanding we start measuring these categories (sometimes splitting them, like colour into intensities at different wavelength). From these more formal variables we can derive other variables that are more abstract, like energy or moments of inertia.

In practice motion seems to be a category our brains are hardwired for: we have neurons sensitive to how things move in our vision, and area V5 is necessary for motion perception. But perceived motion is not velocity: velocity is what happens when you refine the notions of motion by treating it as displacement over time. To some this may be an intuitive refinement, to others it is not.

What is real here? A strict empiricist will say that only what you sense is real. A sceptic may say what you sense may anyway be illusory or at least filtered, so we have very little access to the real world. An idealist would say the abstract notion of velocity is the real thing and your perception of motion a lousy copy. A pragmatic will note that for practical purposes treating things like voltage or momentum as real things works, so quibbling about their reality is a job for philosophers, not physicists.

The refined concept of velocity is not trivial, although most of us usually conveniently forget it. Whether motion always happens relative to other objects or is something absolute lies at the root of relativity theory. The simple "distance divided by time" turns out to be rather complicated when thinking about spacetime. It is not a bad thing as a physicist to sometimes note the weirdness and assumptions underlying the concepts we use.

My interpretation of the question is: is velocity something that really exists in nature or is it just a mathematical construct that we use for calculations?

In classical physics the distinction between reality and math is indeed somewhat blurred, and gives a taste of philosophy. However, from the quantum mechanical standpoint the distinction is very "real": real is what is measurable, and not every quantity appearing in quantum mechanics is measurable. In QM we can define velocity operator and measure the corresponding averages. One case where it is commonly done is measuring electric current, which can be defined as an average electron velocity.

Another aspect mentioned in the OP is the momentum conservation. This conservation has to do with the fact that momentum is one of the integrals of the equations of motion. In classical mechanics there are only seven integrals of motion: three components of momentum, three components of the angular momentum, and energy. What si real is by no means limited to the integrals of motion, so it is a not good criteria for reality.

Finally, it is worth mentioning that physicists use some words of a language in a very restricted sense, whereas life is not limited to physics only. As a teacher of mine said long ago: There is more to velocity than $dv/dt$. By limiting it to being "just $dv/dt$" one robs velocity of most of its meaning.

This is a very deep question. Wow...its really making me think

It seems like you are asking something that, at a deep level, goes down to what is the structure of the universe, i.e. what comprises time and what comprises position, and is there something the independently comprises velocity, or is that just a combination of the components of time and position? No idea what I'm really refering to there. I think I remember quantum field theory...might touch on the subject, but I haven't studied it.

What I do know about that is that, while quantum mechanics (which I have studied a bit in college) describes position & momentum and energy states of atomic particles, quantum field theory describes the constituents of electomagnetic fields

Now, one example of electromagnetic field is light, and so quantum field theory describes light particles (photons)

Whenever an object is in motion, it has to absorb energy to go into motion. Objects in macroscopic world are solid largely because of the electrical forces at the atomic level. And so when you push on the object, there is a cascade of electric (and mangetic?) fields from electrons of atoms pushing on each other to produce the force. I think this involves absorption of photons, i.e. a cascade of absorption of photons, which produces the desired effect

But then you're still left with the same problem: It results in a quantum mechanical description of velocity, which is basically the same thing but couched in terms of probability distributions of both where it is and what its momentum is

I think thats the key thing, is momentum. Is that an independent quality that results in velocity? What is the structure of energy itself?

I suspect that a unified theory that describes matter and energy as separate instances of a more fundamental construct that is common to both would probably shed light on this question. I don't know if that is in any verified theory, but I wonder if the matter energy equivalence from E=Mc^2would be part of it

Will have to look into it when I'm not so damn tired!

Knowledge is a veritable treasure for man, and a source of glory, of bounty, of joy, of exaltation, of cheer and gladness unto him -Baha'u'llah

As far as I can tell, your questions both related to intuition. For instance, velocity sure feels "real" to me, and I think most would agree. That said, there are plenty of quantities - and other mathematical objects - in physics that start to feel less physical and more like mathematical contraptions. Energy is fairly abstract, Lagrangians and Hamiltonians are a step more abstract, and you can go on. Where you draw the line is up to you as long as we can all agree on their definitions and verifiable properties.

I wouldn't get too hung up on this kind of question. It certainly has value, but you can think yourself in circles without getting anywhere. I think physicists, on the by and large, work way too hard at separating out what is "physics" and what is "mathematics" to their detriment. Moreover, they often end up giving a lot of material developed by mathematicians to the physics category (not that us mathematicians are bitter ... ).

Regarding your second question, the short answer is Noether's theorem (something all physicists should be familiar with). For a more intuitively approach, consider the following situation:

a 5 kg ball collides with a 1 kg ball

they were going in opposite directions at 1 m/s

how fast are they going after the collision?

I didn't give you enough information to actually solve this, but I bet you can guess the 1 kg ball is moving faster than the 5 kg ball. In fact, if you add up their velocities before and after, there is likely a net gain. How can this be? Well, some of that velocity is moving around 5 kg of stuff and the other only 1 kg of stuff. It doesn't really make sense to count them the same way ... maybe the velocity that's moving 5 kg around should be weighted more? That's momentum in a nutshell.

I hope this helps.

Some concepts are immediate (or primitive or elementary) in the sense that it is hard to define them. On the other hand we know what they are. That is the case of displacement or time.

Many other are definitions, formed by other concepts. Velocity is defined using displacement and time. We can think of other definitions as kinetic energy, or angular momentum. But they are not real just because are not immediate.

It is like geometry, with point, line and plane (primitives) and triangles or elipses that are definitions.

If in your mind momentum is an intrinsically real quantity, then just take any moving object and divide it into chunks of 1 kg each. If it was an asteroid moving through space, it would keep moving exactly the way it was before without noticing that something sliced through it a lot of times. And the amount of momentum per 1 kg of mass is as good a definition of velocity as any.