During a recent Home Run Derby, Aaron Judge did something that no one thought was possible. He took a swing and hit a ball so hard that it collided with the ceiling at Marlins Park. The ball hit the ceiling about 170 feet above the ground. The height of the ceiling had been designed by engineers so that balls wouldn't hit it—but clearly, they can.

OK, I don't really want to talk about sports. I want to talk about physics. Just how would you even calculate the height of a baseball's trajectory? I'm not just going to show you how to do it, I'm going to let you do it too.

Force and Momentum

I'm going to start with the most important physics idea needed for the trajectory of a baseball: the momentum principle. This says that the total force on an object is equal to the time rate of change of the momentum. Momentum is the product of mass and velocity; both it and the force are vectors.

If you know the forces on an object, you can find its change in momentum. With the momentum, you get the velocity and then can find the new position. That's basically how it works.

Two Forces on a Baseball

After a baseball is hit by the bat, it only has two forces on it (OK, approximately two forces. The first is the gravitational force, a downward force that depends on the mass of the object and the value of the gravitational field (g = 9.8 N/kg). The second force on the ball is a little more complicated: It's the air resistance force.

Although you don't think about it much, you've felt this air resistance force before. When you stick your hand out of a moving window or when you ride on a bike you can feel the force as you move through the air. One of the simplest models for this force uses the following equation: