The Connection Between Muscle Mass and Bat Speed

• Alan M. Nathan's analysis introduces a model where bat speed (v) depends on both bat mass (m) and the batter's mass (M), particularly the muscle mass fraction. This relationship is mathematically represented by (v = k \sqrt{\frac{fM}{m + \epsilon^2M}}), showcasing how muscle mass directly influences swing speed.
• A batter's energy contribution to the bat's kinetic energy is suggested to be directly proportional to their muscle mass, impacting the swing speed significantly. For instance, a batter with a higher muscle mass proportion will inherently be able to swing the bat faster due to greater energy transfer.

Experimental Insights and Model Validation

• A key experiment by Crisco and Greenwald using high-speed motion capture in a batting cage provides empirical support for the model. Data from this experiment indicates a specific exponent value (n=0.28) when correlating bat speed with bat and batter mass, confirming the model's theory up to a certain mass range.
• The model's predictions and empirical data suggest that as the batter's muscle mass increases, so does the swing speed, albeit non-linearly. This finding is particularly pertinent in discussions on the potential effects of PEDs on bat speed and subsequent home run rates in baseball.

PEDs’ Influence on Bat Speed and Home Run Production

• Nathan calculates the effect of an increase in muscle mass on swing speed, finding that a 10% increase in muscle mass can lead to approximately a 3.8% increase in swing speed. This quantification directly ties muscle mass (which can be augmented by PEDs) to a measurable increase in bat speed, offering a physical rationale behind PEDs' potential to enhance home run production.