To complement the baseball-pitching machines at the sports complex where you work, you have designed a catching machine using a mitt, a spring, and a latch that locks the spring in place when the ball has stopped moving. Your boss wants to know the speed of the baseballs coming out of a pitching machine that's been breaking down lately, and you figure you can use the compression of the spring in your catching machine to determine this speed. The mitt that catches the balls has three times the inertia of one ball. Derive an expression for the speed of a thrown ball in terms of the ball's inertia, the distance the spring is compressed, and the spring constant. Take the catch to be a totally inelastic collision between mitt and ball.