# [Physics] Dodging paintballs

estimationhomework-and-exerciseskinematicsnewtonian-mechanicsprojectile

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.

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.