The Physics of Baseball 3: "Throwing Strikes and Breaking Bones"
I recently returned from a conference in Europe, where all of the continent is busily anticipating this year's installment of the Tour de France. It is not only the world's physically demanding race, but is also the finest and most elegant example of raw physics at work in modern sport. The bicycle -- the simplicity of its operation, the gear-ratio dependent torque of the pedals with respect to its drive train, the affect of the decelerating force of gravity as the muscled riders struggle to climb the 6.9% average grade of the Col du Galibier -- is the ideal athletic manifiestion of the poetry of Newton's Laws of Motion as they apply to the science of athletics.
Unfortunately, here in America, professional cycling is less popular than cold fusion and therefore it is pointless for me to waste my time writing about that great sport. Luckily, the art of throwing a baseball presents us with a representation of the poetry of physics that is nearly as beautiful as the art of pedal-pushing. Both involve the application of exteme force to an inanimate object through the use of a bodily appendage. When a baseball is gripped tightly and the arm is thrust forward in a robust throwing motion, it is possible to impart significant momentum to the ball and propel it forward at speeds up to 44.7 meters/second, or 100 miles/hour in layman's units. A related problem involves moving an empty hand (or foot) at high velocity toward a very large object in an attempt to impart momentum to it despite its considerable mass. This is precisely the situation encountered by Major League pitchers Kenny Rogers and Oliver Perez during this past week.
By now, you have no doubt heard about the interaction between Rogers' hand and a water cooler, as well as the one between Perez's foot and a laundry cart . In fact, such instances are nothing new to the game of baseball, as evidenced most recently by Kevin Brown's well-publicized interaction with a clubhouse wall in the middle of last year's pennant race. Let us consider the physical kinetics and dynamics that were at play in an attempt to assign a measure of risk to the actions of both Rogers and Perez.
Rogers' pitches travel at speeds approaching 40 meters/second. However, it would be presumptuous to assume that Rogers' hand is capable of throwing a punch at exactly that speed. For instance, in this case he punched with his non-throwing hand, which is surely slower and less coordinated than the hand with which he pitches. Furthermore, he was not throwing this punch from a mound, which further decreased his maximum attainable punch velocity. Using reductions of 38% for the non-throwing hand and 22% for the lack of a punching mound, we can decuce that Rogers' punch travelled at a maximum speed of 19.3 meters/second.
My unpublished 2003 study of the coefficient of restitution of MLB water coolers revealed that the average clubhouse water cooler is very hard indeed. When striking such an object (typical mass = 30 kg), there is very little "give" to the surface of the cooler. Unfortunately, I am unaware of a similar study of laundry carts in typical MLB clubhouses. However, I am quite certain that the mass of said carts, particularly when filled with the large daily allowance of laundry that is expected from a baseball club, exceed that of the water coolers. Such carts are likely constructed from plastic or metal of a similar hardness to that of the water coolers. Therefore, for the purposes of this calculation, we can approximate both the laundry cart and the water cooler as spheres with coefficients of restitution less than 0.1. This spherical approximation simplifies matters quite considerably, and greatly reduces the complexity of the calculations. Similarly, we can also approximate Rogers' hand and Perez's foot as spheres. Thus, summarizing our approximations diagrammatically, we have:
It should be obvious that the masses of the cart and cooler are considerably larger than those of the hand and foot. Along with the very low "give" of these materials, we can easily solve the conservation of momentum and energy equations. Well, it is easy for me and anyone else with a university physics degree. For those of you who have not had this honor bestowed upon you, you simply need to trust my intellect on this matter. The end result is that, unsurprisingly, the cart/cooler barely moves when struck. The collision is highly inelastic, with a considerable amount of impact energy being absorbed by the fist or foot.
Armed with this information, we can now investigate the possibility of physical correlations between punching a water cooler and the risk of serious injury. A fist speed of 19.3 meters/second results in an impact force of 17 000 Newtons. Assuming that the cross-sectional area of Kenny Rogers' fist is equal to that of my own, we find that the pressure exerted during impact is 2.57 megaPascals. This is considerably larger than the 1.2 MPa compression fracture threshold for bones in the human hand. As for the big toe, its higher volume to surface area ratio (in comparison to the wee pinky finger of the hand) suggests a higher fracture threshold. However, the act of kicking produces lower cross-sectional impact areas than the pugility of punching, and therefore the impact pressure is expected to be higher, probably in the 4.0 - 4.5 MPa range and well above the predicted fracture threshold for toe bones in the human foot.
In simpler, more direct terms, we can conclusively state the following: punching a water cooler hurts like a b*tch. The same is true of kicking a laundry cart. We can furthermore conclusively state that both Rogers and Perez are douchebags. Clearly they had nothing to gain through their actions, for both the laws of physics and the principles of common sense were working strongly against them. In the future, big league pitchers would do well to consult with local physicists before attempting such acts of thuggery.
As always, I look forward to revisiting this blog and enlightening your understanding of our national pastime through the magic of physics. Until then, please join with me in rooting for Der Kaiser, Jan Ullrich. The Tour de France is ready for a new King!