If you’re doing physics only by textbook, the ideal way to catch a ball is to calculate where it will land and then move to that location. I’m going to sort of cheat here, since this calculation of range has already been solved many times. In short, if you know the launch angle (we do) and the launch velocity (yup), then the distance traveled along a horizontal surface would be:
Yes, the most common way to write this equation is to use the double angle formula that gives sin(2θ)—but I included the more basic way of writing it since it doesn’t use a trigonometry trick. Anyway, that’s the location where the ball will land, so the player just needs to move from xp to R during the time the ball is in the air.
Just for fun, here is a model of what that would look like. I’m going to assume the player can instantly determine the initial velocity and angle of the launched ball and then moves at a constant velocity of 5 meters per second to the landing location. Yes, you can have the Python code for this if you need it.
Let’s be clear—baseball players don’t do this. They don’t just move like robots to the correct spot and automatically catch the ball.
Well, then … how do they do it?
Catching a Ball the Human Way
Some people call this the Chapman strategy. It’s from a 1968 paper by Seville Chapman in the American Journal of Physics titled “Catching a Baseball.” Chapman’s idea is that the outfielder will see the ball in the air and then move in a manner such that the apparent position of the ball (with respect to the player) has a constant velocity. This is also called the optical acceleration cancellation (OAC) method. (See: “Catching Fly Balls: A Simulation Study of the Chapman Strategy.”)
But what does OAC really mean? It means that a real-life player is relying on their eyes to figure out where the ball is in relation to them, how quickly it’s moving, and whether they’ll need to back up, scoot forward, or stay put to catch it.