The Rules of the Baseball Press Box
nytimes.com
nytimes.com
Like the author, I was also the kid at ball games who always wanted a ball. I frequent my hometown team's games as a season-ticket holder. I don't always get a foul ball, but when I do, I give it to a kid.
When I don't get a ball naturally, I go to the vendor that sells authentic merchandise (see: genuine) and buy a used game ball. It runs me about $30. It comes with a sticker on it to verify its authenticity. I peel sticker off and rub away the residue. I'll then find a kid who's been working for a foul ball all game and toss them the ball.
They'll celebrate as if they won the lottery. Their parent will usually come find me and say how I didn't have to do that, that their kid is so happy; I joke to them "in 15 years, tell them I bought the ball to give it to them because I knew how badly they wanted the ball."
https://baseballsavant.mlb.com/statcast_leaderboard
The "Exit Velocity" section lists recorded speeds of batted balls broken down by batter, showing their maximum and average speeds. Notice there are a lot of players for whom the average ball they hit comes off the bat at 90+ mph. The "Hard Hit" column lists how many times each player has hit a ball that came off the bat at 95+ mph.
Keep in mind also that balls are also thrown quite hard in baseball, and an important part of a fielder's job is catching balls thrown by teammates.