It's a common sight in action movies where one of the characters is carrying a bow and arrow. At some point he'll fire an arrow but the target simply snatches it out of the air. It looks fantastic and can add quite some drama to any fight scene. But how fast would he have to react in order to perform this trick?
Ok, ok, let's get this one out there first and foremost: don't ever try to catch an arrow with your bare hands, ever! It's stupid, horribly dangerous and can result in death.
That said; let's look at the bow we'll use. There are a lot of different kinds of bows. At the top end is the compound bow. It uses a pulley system to maximize the amount of energy that is stored in the bow when it is drawn. An arrow fired from a compound bow will travel at up to 110 m/s. This means that if the archer fires an arrow at a distance of 110 meters, you'd have a full second to react. That means you have a full second to see the arrow getting fired, have your brain send a signal to your arm to try and catch the arrow and move your arm accordingly.
Suppose you need to move your hand over a distance of about 60 cm in order to catch the arrow. Your hand would have to travel 0.6 m/s or 2.16 km/h in order to catch the arrow before it makes its way through your head. You'll need a little time to react though, about 215 ms. That actually leaves you with 785 ms to catch the arrow, which means your hand would have to move about 0.76 m/s. That's a lightning speed of almost 3 km/h, about the same as the speed of a casual stroll.
Ok, that sounds perfectly doable. But wait; don't go call your archery buddies to go out arrow catching just yet. Think about it, the archer has to be 110 meters away for these numbers. Archery competitions are at a closer range (the longest distance is about 91 meters). Archery isn't the easiest sport on the planet so shooting an arrow accurately over a distance of 110 meters is going to be very hard.
Also, just getting your hand in place isn't enough unless you plan on catching the arrow by having it impale your hand. If the arrow is 1 meter long, you have a window of 9 ms in which to actually grasp the arrow. After these 9 ms the arrow will have passed your hand.
So, it sounds as if it's kind of possible as long as you use a bow and arrow that move slowly. Still, it sounds like a very stupid idea regardless. There are people who perform this trick, though. The Internet is filled with videos of people catching arrows. They usually don't use a high-speed compound bow and quite often they use arrows with a lower speed. Still, this is one trick I won't be trying out any time soon.