How the Physics of Baseball Works

Our discussion of the bat-ball collision has been very ball-centric so far. Another important measure is something known as collision efficiency, or q, a value related to a bat's ability to turn an incoming pitch into a solid hit. To calculate collision efficiency, physicists must consider both the ball's COR and the bat's COR, a value they call ball-bat coefficient of restitution, often abbreviated BBCOR. When they do, they get values for q in the range of 0.2 to 0.25. This in turn can be used to calculate batted ball speed, or BBS, via the following equation:

BBS = (q)(pitch speed) + (1+q)(bat speed)

Plug in some typical numbers, and you get this:

BBS = (0.2)(94 miles per hour) + (1.2)(70 miles per hour) = 18.8 miles per hour + 84 miles per hour = 102.8 miles per hour

A "hotter" bat would have a higher q -- let's say 0.23 -- and would result in a higher BBS:

BBS = (0.23)(94 miles per hour) + (1.23)(70 miles per hour) = 21.62 miles per hour + 86.1 miles per hour = 107.72 miles per hour.

Batted ball speed has a direct impact on how far the ball travels. Increase BBS, and you put a fly ball closer to the fence. By relating BBS to known quantities, this equation reveals something important about the bat-ball collision: that bat speed matters more than pitch speed, but that the bat itself -- how it interacts with the ball -- plays a key role. Wooden and aluminum bats, for example, behave quite differently when they strike a ball. Both types of bats vibrate at the moment of impact, but wooden bats do so in one direction only -- along their length. These low-frequency bending vibrations dissipate much of the energy associated with the bat-ball collision, which means wooden bats don't return as much energy to the ball.

Aluminum bats vibrate in two directions -- along their length and radially as the metal shell squeezes in and then contracts out. This second class of vibrations occurs in a set of frequencies known as hoop modes. The fundamental frequency, or first hoop mode, acts like a spring during collision, compressing in and then expanding out and returning a large amount of energy to the ball. This is known as the "trampoline effect," and it's one reason why aluminum bats lead to higher batted ball speeds. It's also why aluminum bats ping when they strike the ball. The first hoop mode, with a frequency of 1,428 hertz, is associated with a tone that can be heard by the human ear [source: Russell].