In the series September says that he and the ones like him are just humans from the future that could take control over time and go to any time they want to go. But if they are just humans (even it may be possible that over time humans can get more advanced in body and mind) How could they catch bullets that were just shot from a gun? It sounds illogical to me. Anyone has a logical answer to this?
Their technology allows them to move "super-fast" in short intervals, by accelerating their experience of time. This does two things:
Allows them to be fast enough to catch the bullets. The anti-observer guns (first seen near the beginning of 4x22, Brave New World, Part 2) were simply said to "shoot faster".
Bullets are lethal in part due to the physics behind impulse: (Impulse) = (Force) * (Time interval)
Impulse is the change in momentum. The bullet is going from full speed to stopped either when the Observer is hit, or when the Observer catches it; so we can call it a constant in this case.
Force is the force imparted by the bullet on the hand or body.
Due to the acceleration of time, the Observer is increasing the duration of the impact against his hand. Because impulse is constant, that means force imparted on his hand is reduced. To toss in some random numbers:
I = F * t
A) 1 = 10 * 0.1 (Letting himself be hit)
B) 1 = 1 * 1 (Catching a bullet)
C) 10 = 100 * 0.1 (Anti-Observer gun: Letting himself be hit)
D) 10 = 10 * 1 (Anti-Observer gun: Attempt to catch)
A) 1/10th of a second with a force of 10 results in an impulse of 1. Lets call this a normal bullet impact. A force of, say, 5 would pierce the skin and hurt the Observer. If they didn't catch the bullet, and instead let it hit them (as happened mid-series), this could be lethal.
B) When the Observer increasing the time interval be speeding himself up and catching the bullet, force imparted to his hand is reduced. By dropping below the threshold to pierce the skin, it no longer damages the Observer.
C) The anti-observer guns shoot faster, giving them a higher force of impact and a higher impulse. So this is, for example, what the math might look like for one of those guns if the Observer let himself be hit.
D) So even if the Observer was fast enough to catch it, the bullet from an Anti-Observer gun would go through his hand (Well, should go through his hand. As far as I recall, no Observer has ever caught one of these bullets). As it is, when the bullet hits their chest, it's still imparting enough force to hurt/kill the Observer.
Note 1: There are two possible explanations for the Anti-Observer guns hurting the Observer, based solely on the above. Due to events in Season 5, we know that this ability is technology-aided, not a natural ability of the Observers. The other possibility is that this is a limitation of their technology, and they must set parameters before the gun is fired. The bullet would then speed past their hand because they cannot catch it, but before it impacts them, the Observer drops back to normal time and case C is the result.
Note 2: Caveat - the numbers are made up, not calculated from real life physics, it's just a simple, easy-to-follow example.