Saturday, April 19, 2014

Can a baseball pitch ever travel faster than the pitcher's hand?

From Bjorn the Berzerker comes this baseball question:

Will the ball ever travel faster than the speed of the pitcher's hand?

My off-the-cuff answer was that it could, because my perception was that the pitcher normally throws somewhat "down" at the batter. When the ball's motion has a downward component, gravity acts in the direction that speeds it up. The extreme case is if you just lobbed it straight down -- gravity would accelerate the ball faster than you threw it, until it hit the ground. If you're throwing a ball "down" from the mound to the strike zone, it seems plausible that gravity plus the downforce on a good curveball could well outweigh the slowdown due to air drag, even taking into consideration that air drag directly opposes the motion while downward forces are nearly perpendicular to the motion. (Remember that the speed of the ball is the vector sum of its horizontal and vertical velocities.) 

But on further consideration, it doesn't work out that way. Pitchers aren't throwing down at all, as I'll prove below. In my defense, judging slight deviations from horizontal is very difficult, as anyone who's ever been to the Mystery Spot can attest.

For simplicity, consider a perfect 12-6 curveball that stays completely within a vertical plane all the way to the catcher's mitt. If the pitcher is six feet tall and throws overhand from the standard 10-inch mound, the ball leaves his hand about 82 inches above the plate. I'm assuming here that the release point is six feet above the ground. (Pitchers release the ball over their heads, but they're not standing up straight when they do so.)

I'll assume the pitch has to be a strike, because otherwise the pitcher could just throw the ball straight down at the ground, which we have already discussed. The lower limit of the MLB strike zone is the bottom of the batter's knees, about 24 inches off the ground. So the pitcher has 82 - 24 = 58 inches of drop to work with. 

Let's say the pitch is an 80-mph curveball. Pitch speeds are measured (ideally) at the release point. This expert says pitches lose about 10% of their speed due to air drag by the time they get to home plate, so our hypothetical pitch would be going 72 mph at the plate. 

The distance from the pitcher's hand to home plate is about 58 feet or 696 inches. (It's 60.5 feet from the pitcher's rubber to the plate, but the ball is released when the pitcher is stretched way out in front of the rubber.) A ball moving at an average of 76 mph will take 0.52 seconds to cover the 696 inches to the plate. In that same 0.52 seconds, gravity will cause an object released with no vertical speed to drop 52 inches.  

That takes up almost all of the available 58-inch drop! It means that even a fastball has to be thrown nearly horizontal in order to be a strike. A downward-breaking curve would have to be thrown slightly above the horizontal.

This means there is no way the downforce could cause the ball's speed to increase after release. A horizontally thrown, 95-mph fastball takes 0.43 seconds to get to the plate and gains 9.6 mph in downward speed over that span of time (actually a little less, because fastballs have a slight upward break, but neglect that.) If the fastball loses 10% of its speed over the 0.43 seconds, that is an average deceleration of just about one g. There is no way the small fraction of the one-g gravitational acceleration acting along the ball's nearly horizontal path, can overcome that. (The one-g deceleration is perfectly consistent with this guy who says the terminal velocity of a baseball is 95 mph.)

It is even less likely that gravity could speed up a curveball, because the bigger your curve breaks, the higher above horizontal you have to throw it in order to hit the strike zone, and the later in the trajectory it reaches its apex and starts falling. By then, air drag has cut down on the speed so much that no reasonable amount of downforce is going to bring it back up to its initial speed. 


  1. Thank you. What if we remove air drag and gravity? I'm being a bit basic. I can't seem to grasp if the pitcher's hand/arm is accelerating with the ball, will the ball continue to accelerate, even for just a moment, after leaving the pitcher's hand. I've always thought that the ball could not go faster than the pitcher's hand.

  2. Acceleration can only be caused by a force. Once the ball leaves the hand, the only forces that can act on the ball are air pressure/drag and gravity. So without those, the ball cannot accelerate after leaving the pitcher's hand.

    There is one more thing to say. The rotation of the Earth affects the motion of the ball relative to the ground just like a force would. This is called the Coriolis effect, it varies by latitude, and it can move the ball laterally or up-and-down depending on the direction of the pitch. It is a small effect, down in the submillimeter range.

    A 1 mm height difference may seem significant when you consider round bat versus round ball, but you have seen those high-speed photographs showing the ball squashing down as it contacts the bat, so 1 mm is probably on the very lower bound of significance.