Who Really Supplies The Power: The Pitcher or The Hitter?

 I always grew up hearing this from coaches when a power pitcher was on the mound:

"He'll supply the power. Just make contact."

It seemed to make sense. The faster the ball hits the bat the further it will travel after impact.  Except the science says otherwise.

The formula to determine the speed at which the ball rebounds off the bat is:

                                                           v_exit = q v_pitch + (1 + q) v_bat

EXIT VELOCITY (OFF BAT) = ea * PITCH VELOCITY + 

(1 + ea) * SWING SPEED

Where v_exit  is the ball exit velocity, v_pitch is the velocity of the pitched ball and is the v_bat  speed of the bat.  "q" is COLLISION EFFICIENCY

Collision efficiencyvalue is between 1.0 and 2.0 depending on the squareness  of the hit. A well hit ball with a wood bat on the sweetspot gives a value of 2.0.   A detailed explanation of COLLISION EFFICIENCY is in this paper.  I used a value of 1.5 for q with graphs below.

So what does this formula tell us?  It shows mathematically that the batter's swing speed is 6 times more important than the pitch speed in creating higher exit velocity!

Graphs below show the exit velocities for 5 several bat speeds (60, 65,70,75,80 mph) as pitch speed increases from 65 to 100 mph. 

Notice how flat the exit velocity increases are for each bat speed. While there is an increase as the pitch speeds increase (pitch speed does have an influence) it's not a large one.

When we increase both the pitch speed AND the bat speed, the exit velocity increases more sharply.

So for a pitch speed of 65 mph and a relatively show bat speed of 60 mph, the exit velocity after a decently hit ball would be around 79 mph. An increase the pitch speed to 85-90 mph with a commensurate increase in bat speed to 70-75 mph would yield an exit velocity above 95 mph. Similarly a bat swung at 80 mph making contact with a 95 mph pitch would cause the ball to leave the bat at 105+ mph.  

Bat speed not pitch speed supplies most of the power!