To examine a softball pitch with a physicist's eye, I enlisted the help of my experimental colleague Will Roach. We set up a camera to film Lynchburg College softball pitcher Hope Johnson during her warm-up in the bullpen. The animated clip below shows Hope firing a two-seam fastball.
Blogger won't let me upload the animated GIF, so I went for the movie format instead. I've identified two tracks in the video. The one in yellow shows the path of the softball. The one in red shows the path of Hope's shoulder. The reason for the second path is to use the shoulder as a reference frame for rotations about the shoulder.
There is so much great physics in that video! Hope begins with the ball in her mitt. She then rocks back slightly as she pulls the ball out of her glove. Note early in the video that her back (left) shoe has the toes off the dirt. The soft rocking generates a little momentum that she uses as she initiates the full motion of the pitch.
A little later in the clip, Hope is moving forward at about 3 m/s (11 kph or 6.7 mph). Note her front (right) foot pushes off the pitcher's rubber. That added push helps Hope's translational, or forward, speed reach about 4.9 m/s (18 kph or or 11 mph) at its maximum, which happens just as her front foot plants into the dirt. At that point, her linear momentum shifts to angular momentum as her hips and torso turn during the point of release for the ball. The graph below shows the progression of the speed of Hope's shoulder as measured by a ground observer (click on the image for a larger view).
Hope's translational speed obviously helps increase the speed of her fastball. The graph below shows the progression of the speed of the softball as the pitch unfolds (click on the image for a larger view).
The speed of the ball at release is about 27.8 m/s (100 kph or 62.3 mph). The drop in speed after release is due to air resistance. Pitch speed is further increased by cocking the wrist back, thus storing potential energy. At the point of release, the hand flips forward like a spring, releasing some of that stored energy.
We know in physics that the net work on an object is the change in the object's kinetic energy. Given that work is, qualitatively, a force times a displacement, increasing the distance over which a force acts is one way of increasing kinetic energy and, hence, speed. That is the beauty of the loop-the-loop pitching motion! Look at the photo below (click on the image for a larger view).
By executing the loop-the-loop pitching maneuver, Hope is able to work on the ball over a much larger distance than the forward distance her shoulder moves. The yellow path in the video and in the above photo shows that the ball must undergo centripetal acceleration as part of its total acceleration. The graph below shows the magnitude of the ball's acceleration as measured by a ground observer (click on the image for a larger view).
Coming through the bottom of her 70-cm (28-in) right arm's pendulum-like swing, the ball has an acceleration of nearly 47 times the acceleration due to gravity! Her right arm must exert a force on the ball of over 88 N (20 lb) at the bottom of the swing, just before the release point. That force is 25% more than the weight of the heaviest bowling ball!
During the swing of the arm, the ball rotates about Hope's right shoulder, in addition to the translational motion it has because Hope is moving forward. The graph below shows the angular speed of the ball as measured in the rest frame of the shoulder (click on the image for a larger view).
The ball's maximum rotational speed is about 30.3 rad/s or about 289 rpm. That rotational speed is about two-thirds the rotational speed of helicopter blades!
The last graph below shows the ball's kinetic energy during the pitch sequence (click on the image for a larger view).
The peak kinetic energy at the release point is nearly 73 J (54 foot pounds). That size kinetic energy is not something you want hitting your unprotected head. For comparison, a bowling ball would have that same kinetic energy if it had a speed of about 4.5 m/s (16 kph or 10 mph). I definitely don't want a bowling ball hitting my head at 10 mph!
For all the great physics discussed above, a lot more remains untouched. I've not gotten into the biomechanics of the arm and leg movements. I've not discussed energy in the body, friction with the ground dirt, ball grip, and a whole host of other fun topics. For now, watch the video again and enjoy the beautiful loop-the-loop!