Dodging paintballs I was recently watching The Matrix, and was considering the speed at which one needs to move to literally dodge a projectile. Now, in real life, it is impossible to dodge bullets as such a feat would require superhuman reflexes and knowing exactly where the bullet will hit. 
However, paintballs, which are much slower than any bullets, might be another thing. I was wondering about the fastest paintball speed, which is probably lower than 160 $m/s$. The legal limit is set at about 90 $m/s$, but I am sure that it can go a bit faster to that with a "Mythbuster-type" custom paintball marker. 
If I assume that I am about 8 m from a person, and that person needs to move at most .50 m to dodge my paintball, what would have to be my target's speed in order to barely dodge that paintball?
Note: Here, air resistance is already taken into account for the speed of the paintball.
 A: Correct me if I'm wrong, I believe that from a paintball gun when shooting there is no smoke, vapour or other visible exhaust and not even recoil. No visible signal. The dodger must rely on sound.


*

*Sound reaching the ears. The speed of sound in air is around $v=330\:\mathrm{m/s}$. Sound will propagate the $x=8\:\mathrm{m}$ and reach the dodger in $$t=x/v=0.0242\:\mathrm{s}$$

*Start signal reaching the muscles. The reaction time can be defined as the time getting the signal from the brain to the feet. This source finds that time to be $0.026\:\mathrm{s}$ for a person of average height.
As a remark, in the Olympics a starting time from startshot to first reaction (the sum of the two values above) below $100\:\mathrm{ms}=0.1\:\mathrm{s}$ is considered a false start. This is deemed not humanly possible. http://condellpark.com/kd/reactiontime.htm 
Your actual question what would have to be my target's speed in order to barely dodge that paintball? is not really dodging. If I dodge something, I will start by standing still and then accelerate away. I will not have a constant speed from start to end of my dodging. Therefore:


*

*Dodging. This source gives the acceleration of Olympics recordholder Usain Bolt at a $100\:\mathrm{m}$ sprint at the value $a=9.5\:\mathrm{m/s^2}$. Assuming Usain Bolt is the dodger and can keep this acceleration constant, the time to move half a meter is: 
$$x=x_0+v_0t+\frac{1}{2}at^2\implies t=\sqrt{\frac{2x}{a}}=0.324\:\mathrm{s}$$


As @Victor's answer shows, the bullet will hit after $0.05\:\mathrm{s}$. So not even starting to dodge is possible, and dodging alone is by far impossible even if you started exactly when the shot is fired.
We could try to alter our values though. For instance, half a meter is not necessary, the distance could be larger, there might be visual signals instead so sound propagation time is not interesting, etc.
A: Arrows and paintballs go about the same speed, 200-300 feet per second.  In episode 109, Mythbusters confirmed a professional ninja can block an arrow with a sword.  It's plausible.
However, distance is going to make a difference.  I don't have the episode to check how far away the archer was, but in your question 24 feet (8 meters) is really close.  It only gives about 100 ms of response time (reaction time + movement time).  A human's reaction time to auditory stimulus (the sound of a paintball gun firing) is 100 - 190ms.  So no, a human couldn't dodge a well-aimed paintball if all they have to go on is seeing the paintball.  They could barely start moving at the moment they're hit.
BUT! The gunman will have a response time between aiming and firing. A smart target human will react to seeing the gun pointed at them before the gun fires. The time it takes for the gunman to steady their aim, react and pull the trigger cancels out the target's response time.  By the time the gun fires, the target can already be responding.  This is similar to the effect of opening the distance between the target and gunman.  As Mythbusters has already demonstrated, with more distance an arrow/paintball can be dodged.
In conclusion, no a human cannot dodge a paintball at 8 meters once it is fired.  However it is plausible that a human can dodge a paintball at 8 meters if they notice the gun at the moment it is aimed at them.
A: The time that the person has to dodge is
$ t=\frac{8 m}{160 m/s}=0.05 s $
So he has to move at a minimum speed of
$ v=\frac{0.50 m}{0.05 s}=10m/s $
A: As always in this sort of subject, you should consider Mythbusters, not just mention it. 
https://www.youtube.com/watch?v=2qY0DA7DG9s
And the answer is pretty much, no you can't. It's reaction time that's the problem.
A: I've dodged a paintball from 10-15 feet away before. It was in my friends house at the time and we were messing around but it is possible to do. As far as I remember I was in a room down a hallway looking around the corner of the doorway and saw him down the hall. My just reacted as I saw him pointing the paintball gun in my direction and I pulled my head back just in time to hear the ball hit the wall behind me. So is it possible? Yes.
