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When you hit a baseball or softball, the bat imparts kinetic energy onto the ball, causing it to launch onto the field. However, the ball also imparts energy onto the bat! Some of this energy goes into rotating the bat about a pivot point. Imagine a stick floating in a tub of water. If you push down on the stick near one of its ends, it will rotate. The same thing happens to a bat when it comes into contact with a ball. It tries to rotate and in the process, it pushes against your hands!
But this is only part of why hitting the ball can really sting. Some of the ball's energy also goes into vibrating the bat. When you tap a tuning fork, what happens? You can see (and hear) the prongs of the tuning fork vibrate in response. A baseball bat reacts similarly when it hits a ball. The bat can vibrate at frequencies around 200 Hz, and the vibrations near your hand can sting! Interestingly, if you hit the ball at a particular spot on the bat, the vibrational waves that propagate along the bat will cancel each other out. When this happens, you don't feel any vibrations in your hands (so they don't hurt as much). Furthermore, energy is no longer lost in the form of bat vibrations, so the ball can travel much farther! This spot on the bat is sometimes called the sweet spot.
Have you noticed that on a cold day, hitting a baseball or softball hurts a lot more? This is because colder balls are less elastic, so more energy is transferred into vibrating the bat (instead of deforming the ball). This also explains why rubber balls (like lacrosse balls) hurt a lot less when you hit them with a bat.
So if you want to avoid the stinging pain of hitting a baseball or softball, just make sure to only play on warm days and always hit the ball on the sweet spot of the bat. Easier said than done, right? And they say baseball players aren't tough.
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