Physics Behind a Softball Pitch

If you already miss watching baseball less than a full day since the Red Sox won the World Series, head out to the softball field for a little fun and exercise.  Beware, though, if you get into a fast-pitch softball game.  Swing quickly because the ball will be on you in a hurry!

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!

The Point of Center of Mass

There is a reason so many complicated problems may be analyzed with relatively simple introductory physics.  We in physics sometimes refer to the "spherical cow approximation" when simplifying complicated problems.  The main principle behind the simplification is that an object's center of mass moves under the influence of the net external force acting on the object.  So if we are not interested in all the goofy motion about the center of mass, we need only focus on the translational motion of the center of mass.  The cow is thus approximated by a point particle.

After introducing the aforementioned idea to my introductory physics class this morning, I decided to film one of my favorite objects -- a baseball bat -- in flight.  With the help of my experimentalist  colleague, Will Roach, we made the video below (click on the image for a larger view).

I marked the location of my bat's center of mass and then tossed it across our campus lawn.  The red circles and red line in the clip above show the path of the center of mass.  What does that path look like?  Check out the graph below, which shows the location of the vertical coordinate as a function of time (click on the image for a larger view).

The red circles are the data.  The blue curve is a parabolic fit to the data.  The constant vertical acceleration predicted by the fit is 9.838 m/s² (pointing down), which is almost exactly the magnitude of the acceleration due to gravity of a point particle moving in a vacuum near Earth's surface.  Given that air resistance on my bat was not a big player in my simple toss experiment, I think the above video illustrates rather well the fact that the bat's center of mass follows the path predicted by relatively simple introductory physics.

To understand the more complicated motion about the center of mass, one must study rotations, a topic coming soon to my introductory physics course.

The Power of an Axe Kick!

My last post examined a punch from my karate instructor, Mr Abercrombie.  In this post, I examine his axe kick, which involves bringing the leg up as high as possible, followed by a fast downward kick.  The knee bends very little, which makes it appear as if an axe is swinging downward.  Check out the video below (click on the image for a larger view).

Mr Abercrombie begins by executing what is called a crescent kick.  Instead of swinging across as with a crescent kick, he brings his foot straight down.  Mr Davis holds the board, which acts as victim in this demonstration.  The image below shows Mr Abercrombie's foot at the apex of its motion (click on the image for a larger view).

His foot passes through the board at about 10 m/s (36 kph or 22 mph).  His foot's acceleration through the board is nearly 15 times the acceleration due to gravity.  I included acceleration vectors in the video so that you may see how the magnitude and direction of his foot's acceleration changes during the kick.  You'll note the downward pointing acceleration as the foot rise, which is necessary to slow the foot down.  A great deal of potential energy is stored at the instant seen in the above photo.  Once he brings his foot down, the acceleration begins to turn to the left in the movie.  That is because his foot moves along the arc of a circle during the hit, which means centripetal acceleration must be present.  At the point of contact, his foot delivers about 2200 N (500 lb) force to the board, breaking it easily.

I modeled Mr Abercrombie's leg as a sequence of rigid rods.  Knowing the percentage each segment comprises of a man's body mass, I can get a reasonable estimate of the moment of inertia of Mr Abercrombie's left leg.  During the explosive kick through the board, Mr Abercrombie's leg has a rotational kinetic energy of over 200 J (almost 150 ft lb).  In the roughly one-eighth of a second it takes for his foot to go from rest at the apex to breaking the board, more than 1500 W (just over 2 hp) of power is required.  Though an athlete like Mr Abercrombie is capable of large power outputs for only a very short length of time, I am very impressed that the power output of such a kick exceeds two horsepower!

Given that the body is only about 20% efficient in its energy conversions, Mr Abercrombie probably burned about 1000 J (almost 750 ft lb) internally just to move his leg from its apex to the board.  That amount of energy represents about a quarter of a nutritional Calorie.

Did you ever imagine so much power in a kick?

Karate Hits Physics of Sports!

My Physics of Sports students were treated to something special this morning.  Third-degree black belt Clifton Abercrombie and second-degree black belt Cody Davis visited my class and showed us some karate moves.  Mr Abercrombie has been one of my family's lead karate instructors for about a year-and-a-half now.  He is incredibly good at what he does.  The slow-motion video below shows one of his board-breaking feats.  He executes a perfect hammer punch, which uses the knife edge of the palm opposite the thumb.  Using the edge of one's hand allows for more force per area, i.e pressure, at the strike point.  Click on the video for a larger image.

There is a great deal of physics here!  At the start of Mr Abercrombie's motion, you will see his torso beginning to rotate.  His right arm goes back so that as his torso moves forward, he will store a great deal of potential energy in his upper arm.  Once his strong thigh, abdominal, and back muscles get his torso rotating counterclockwise (as seen from above), his right arm is like a whip.  Now cocked full of potential energy, his arm moves forward, thus turning potential energy into kinetic energy.  The red path in the above video clearly shows that after accelerating linearly in the early part of the motion, Mr Abercrombie's hand undergoes mostly centripetal acceleration as his arm follows through and breaks the board.  Note, too, Mr Abercrombie's eyes and how they are focused on his target.  As he follows through with the hammer throw, you will see his mouth open.  He yells, "Kiai!"  His exhale during the blow demonstrates proper breathing technique.

Let's now get more quantitative.  The graph below shows the speed of Mr Abercrombie's hand.  Click on the image for a larger view.

His hand's maximum speed is just over 12 m/s (43.2 kph or 26.8 mph).  That is speeding in a school zone!  To understand what accelerations are involved in making one's hand follow the path you see in the video, check out the plot below.  Click on the image for a larger view.

His hand's maximum acceleration is just over 325 m/s², which is over 33 times the acceleration due to gravity!  A grown man's arm is nearly 6% of his body weight.  I estimate the mass of Mr Abercrombie's right arm to be 5 kg, which corresponds to a weight of about 11 lbs.  He needed about 1641 N (369 lbs) of force on his right arm to execute the hammer punch.  What's great is that he most likely used more force because my camera films at just 60 frames per second, meaning my data points are separated by 0.0167 s.  With a high-speed camera, I could get many more data points, and, I suspect, get instantaneous forces approaching twice the size of what's seen above.

What to learn to break a board like Mr Abercrombie?  Visit Super Kicks Karate in Forest, Virginia.  Click here for their Facebook page.  It's a great place to see physics in action!