Adding Velocities

An analogy is in order to understand what addition of velocities means to me as an engineer. Let’s assume you stand in the bed of a pickup with the pickup not moving and you throw a baseball 75 miles per hour directly over the cab and hood of the vehicle. Now, let’s say you remain in the bed of the pickup and throw the baseball 75 miles per hour but this time the truck is being driven 15 miles per hour forward (dangerous, I know). Now, the actual baseball velocity will be 90 miles per hour with respect to the ground (75 miles per hour due to you plus 15 miles per hour due to the truck).
An addition of velocities is what I believe the pitcher tries to achieve with his throwing mechanics. The implementation isn’t quite as simple as the pickup analogy, but if the arm can accelerate the ball to 75 MPH without any significant help from the body, and the upper body can rotate the straight arm so the ball goes 15 MPH, the potential is there for the baseball to be thrown at 90 MPH; the addition of the two velocities. To appreciate this addition you may want to refer back to Turbo Effect and the discussion about our experience when Paul “lost” the rotational speed of his lower and upper body after hip surgery and consequently lost 20 MPH of velocity.
Because the ball speed due to upper body rotational speed is much  slower than the arm speed (15 MPH versus 75 MPH in this example), the upper body needs to get a head start and be close to completing rotation before starting the forward action of the arm.
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The assumption here is that to achieve maximum ball speed, both the arm and the body need to be moving  at maximum rotational speed at the same time. This timing is critical and takes a lot of practice to optimize. Ideally, the full external rotation of the arm in the shoulder socket occurs relatively early in the upper body rotation as we’ve discussed previously. When the upper body is about three fourths or more through the quarter turn, the arm acceleration can begin (coming out of external rotation). When the upper body has rotated further and is facing the target, the arm should be at full extension for release of the baseball.
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If these two elements of throwing are timed perfectly, the upper body rotational velocity will be maximum when the arm speed is maximum and the two will be added together.
We like to think about this sequence occurring because the upper body starts by pulling the arm. The upper body rotation externally rotates the arm and then at the optimum time the pitcher fires the shoulder and arm muscles to accelerate the ball. Again, the goal is for the upper body rotation to get to maximum speed at a point where the chest is facing the target.  At the same instant the hand is at full extension and the ball is released to the target.
Paul has informed me that the following section is not going to be understood by too many baseball people, but I wanted to include it as further evidence of the addition of velocities and the required timing related to external rotation of the arm in the shoulder socket and torso rotation.
Timing Calculation (Nerd Warning!)
When should the arm start forward out of full external rotation if the assumption is made that shoulder speed and arm/hand speed can add together to enhance ball speed? The following calculation will provide some insight into the critical timing between the shoulder rotation and the arm rotation to maximize ball speed. First, the assumption is made that the hips will lead the shoulders and provide the “wound spring” effect in the torso so the shoulders can move at maximum speed. Similarly, the assumption is made that the arm externally rotates early in the shoulder socket to maximize the “wound spring” effect in the shoulder.
For the following calculation, we’ll assume that the throwing velocity will be 90 MPH with 75 MPH contributed by the arm/hand and 15 MPH contributed by the torso/arm/hand. These numbers are just an example, but are somewhat representative of the earlier discussion of Paul’s hip injury and the resultant drop in velocity that resulted.
Intuitively, you can look at these two relative velocity numbers and the resultant 5:1 speed ratio and know that the arm needs to start moving forward late in relation to the shoulder rotation if the two throwing elements are going to be going at their maximum speeds simultaneously at release of the baseball. We explain to students that’s it’s like a Beetle and a Corvette that are going to run the quarter mile with the objective to end up at the finish line together. The VW needs a significant head start on the Corvette.
The following is the calculation of the starting point for the arm to come out of external rotation. The calculation uses my arm, back and shoulder measurements.
If the hand is going to be moving 75 MPH at release (due strictly to the arm) and starts at zero MPH, the average velocity will be approximately 37.5 MPH from full external rotation to release. An average speed of 37.5 MPH is 660 inches/sec calculated at (37.5MPH x (5280 ft/mile x 12 in/ft))/(60 x 60 sec/hr). The distance from the center of the baseball to the elbow is 15 inches, so the length of the 90 degree arc that the baseball will travel from full external rotation (zero velocity) to release of the baseball (75 MPH) will be about 24 inches ((2 x pi x 15)/4). The time it will take the hand to make this arc is 0.036 seconds (24 inches/660 inches/sec). Just as a reference, that’s about 1/10th of the time it takes you to blink your eyes.
The further assumption for this calculation is that the torso rotational velocity can add 15 MPH to the release velocity so that’s an average of 7.5 MPH from start to finish of torso rotation. A speed of 7.5 MPH is 132 inches per second, developed with the same type calculation as shown above for the shoulder. The distance from the middle of the back to the elbow is 21 inches so the 90 degree arc the elbow will traverse through torso rotation is about 33 inches ((2 x pi x 21)/4). This arc distance is measured from the point where the shoulder line is aimed at the target to where the shoulders are square to the target at release of the baseball. The time it will take for the torso and elbow to make this 33 inch arc is 0.25 seconds.
Looking at these two arc times shows that the actual ratio of rotation times is about 7:1 for someone like me that is 6′ 1″. The calculation means that the hand takes 1/7th the amount of time to go the 90 degrees from full external rotation to release compared to the time it takes the shoulders to rotate the elbow 90 degrees to get square to the target. Simply stated, the shoulders must be approximately 86 percent of the way to square before the hand starts to accelerate if the shoulders and the hand are going to be synched to achieve maximum velocity at release. An 86 percent completion point is 77 degrees of rotation from start or just 13 degrees from where the shoulders are square to the target.

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These two pictures of Paul would tend to confirm the above calculation. There are only a few degrees of upper body rotation remaining in the first picture with the arm still in full external rotation. In the second picture the shoulders have completed the last few degrees of final upper body rotation while the arm has come out of full external rotation and completed its 90 degree arc path to achieve the full extension release point.
This timing calculation helps explain why it’s so difficult to maintain a constant velocity and how easily someone like Justin Verlander could lose several miles per hour with a stomach muscle issue and especially how Paul could lose 20 miles per hour with a major hip issue. It’s obviously difficult to maintain the same velocity from pitch to pitch with this tight timing requirement and this analysis should make you realize how incredible the human body is for a professional pitcher to be able to repeat the action as optimally as he does. As was discussed earlier, the pitcher can add some folding of the upper body to enhance velocity if the rotational elements are not perfectly matched, but folding is a less efficient throwing motion than pure rotation.