Mosquitoes fly at about two kilometres per hour (1.2mph), which sounds slow, but at their size it's like you flying over a hundred times that! They are going blindingly fast for something that small.
You can't catch them for a few reasons. One, they see you coming. Their eyes are big and round and multifaceted, so they are quite capable of seeing above and behind themselves. You can't sneak up on a mosquito easily.
Second, your hand moves slower than they fly and has more distance to cover. The mosquito doesn't need to move far to get away, while your big and slow hand has to travel all the way to your head.
Lastly, the mosquito makes decisions faster than you. When you decide to hit a fly, a signal goes from your brain to your spinal cord to your arm muscles to start the hand in motion. The time it takes is a few milliseconds. But once the mosquito sees motion, a signal from its brain goes to its nerve cord to its wing muscles, and the time ends up being a fraction of a nanosecond. They think and act 100 times faster than you can. Your hand never really had a chance!
Matan Shelomi, entomologist
To add to Matan's excellent answer, air pressure from your hand also serves to blow the extremely light mosquito out from under it prior to impact, aiding in escape. This is why fly swatters are made of mesh, not a solid panel.
Insects have fast-acting, wired-in escape strategies.
However... someone taught me once how to kill a house fly with your hands, and it works.
The trick: when the fly is sitting on a surface, clap your hands together above the fly. The fly will see the motion and take off going up, flying into your hands just as they come together. Insect reflexes are fast, but also predictable.
False, I can really consistently nail mosquitoes, even in mid-air. But I can never catch those dang flies without a swatter.
They're usually hard to see, and there's too many. Should you kill one, expect another one to be snacking on you in the meantime.
One low-tech solution to mosquitoes is a box fan. Mosquitoes fly poorly as insects go, and if you have a steady breeze going, that seems to keep at least most of them at bay.
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