Reversibility in newtonian mechanics It will be slightly different from related questions. 
When we write newton's differential equation $$\ddot{x}=\frac{d^2x}{dt^2}=m\,a$$
we see that reversing time doesn't change the force, but it does change the velocity. Then we ask: 
Why we don't observe the reverse motion?

What I can't understand is the following: to solve equations we actually need reality. We need velocity and its direction, and the movement is always determined by it. 
What is the strangeness? 

Any book treating this with clear examples will be greatly welcome. 
 A: As you have discussed, there is nothing in Newton's laws to give a preferred direction in time. This is where people usually mention "the arrow of time" that tells us which way time flows. This is just another way to view the second law of thermodynamics: the entropy of an isolated system can never decrease. If we think of the entire universe as an isolated system then, we see that time flows in the direction of increasing entropy. 
The arrow of time actually does break down in two situations. The first is when we have a single or small amount of particles. For example, projectile motion. If you were to see a video of a projectile, you could play it backwards and nothing would be out of the ordinary. In other words, there are too few particles for entropy to be significant, or even definable.
The second example is when our system is at maximum entropy. Imagine a container divided by a partition with one half filled with gas. Once we remove the partition, entropy will increase as gas molecules fill the empty part of the container. We can see time "flowing" here, since it would be odd for us to suddenly see a bunch of gas molecules fill half a container and leave the other half empty (which we would describe as time moving backwards). But once the container is well mixed at maximum entropy, we no longer see a flow in time. If we were to record the molecules for a while after this point, and then reverse the video, once again nothing would be out of the ordinary. We would just see "random" movent of gas molecules.
I bring up these two examples because Newton's laws hold the entire time in both situations, but there is no indication of a "direction of time". We need the second law to tell us this. To go back to the second example, Newton's laws say that it is possible for all of the gas to end up back in just one half of the container. Thermodynamics says it is highly unlikely, which for our finite capabilities means impossible. In other words, Newton says time can flow backwards, while thermodynamics says it cannot. 
A good reference for this is the original idea of the arrow of time found in Eddington's "The Nature of the Physical World" if you want to learn more about this.$^*$

$^*$ This also has one of my favorite quotes:

The law that entropy always increases⎯the second law of thermodynamics⎯holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations⎯then so much the worse for Maxwell’s equations. If it is found to be contradicted by observations⎯well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. 

