Soriano, his Bat, and Momentum

Hitting a baseball is all about momentum. Momentum is a vector describing a "quantity of motion" or in mathematical terms p (momentum) = mass * velocity. In a closed system, momentum is conserved. Conservation of momentum means that the harder you throw the ball, the harder the ball will bounce back. Think in the case of throwing a ball against a solid wall. The harder you throw the ball, the harder it bounces back. This is easy to see in the case that fast balls tend to be hit harder and further than curve balls. What this also means is that the bat transfers momentum to the ball. This is why it can be better to use a heavier bat if you can swing it just as fast. The momentum is a product of the mass and velocity, so a heavier bat swung with the same speed as a lighter bat will have more momentum.
 
But real life is not a closed system, so in the process of hitting a baseball you will lose some momentum. Part of this momentum is lost in the deformation of the ball and to a lesser extent the bat.
 

 
A number to determine what this compression does to the ball is called the Coefficient of Restitution (COR). The COR rating for a baseball is 0.555, which means a ball will return 55.5% as fast as it came in. When I used this number with another known my numbers didn’t come out correct and I think this is due to the COR of a bat. I was able to get the numbers to work if I use a COR for ball + bat of .37.
 
The formula for the balls exit speed will be:

Where:
va is the final velocity of the first object after impact (ball)
vb is the final velocity of the second object after impact (bat)
ua is the initial velocity of the first object before impact (ball)
ub is the initial velocity of the second object before impact  (bat)
ma is the mass of the first object (ball)
mb is the mass of the second object (bat)
Cr is the coefficient of restitution
 
What we can assume is that Alfonso Soriano isn’t swinging the bat as fast due to his aging. He just can’t swing the bat as fast he can when he was younger. Let’s say 6 years ago in 2006 when he had 46 home runs his bat speed was 81 mph*.  We know the weight of his bat was 33 oz. So his momentum is 81 mph * 33 oz = 2673 mph*oz (this is a made up unit, but it will work). Let’s say that over the years his bat speed has dropped 3% to 78.55 mph while still swinging the same bat. The new bat speed momentum is 78.55 mph * 33 oz = 2592 mph*oz. This is going to relate to less momentum being transferred to the ball which means the balls aren’t going fly as far. Soriano may still hit a 95 mph fastball** out of the park if he makes contact with it, but an 85 mph curveball may not have enough momentum to make it out of the park.
 
*I am using 81 mph because I read that Albert Pujols’ bat speed was 86.99 mph with a 31.5 oz bat. This gives us a momentum of 2740 mph  * oz. I’m just going to assume Soriano is his prime was a little worse than that. For reference, it is said that Babe Ruth had a 54 oz bat and swung at an estimated 75 mph.
 
**A baseball has a weight between 5 oz and 5 ¼ oz. This gives an average weight of 5.125. At 95 mph, the ball has a momentum of 486.9 mph*oz. At 85 mph, the ball has a momentum of 435.6 mph*oz.
 
Sveum has asked Soriano to drop his bat weight down an ounce, from 33 ounces to 32 ounces. This is to increase his bat speed. In our previous example I assumed that Soriano had a bat speed of 81 mph. If he returns to this bat speed with the loss of an ounce, his new momentum will be 81 mph * 32 oz =  2592 mph*oz which is where it was with a larger bat at a slower speed. What it looks like though is that the exit speed of the ball will be more, even if he uses a lighter bat, so it should pay off.
 
Another thing to consider is that the change in bat speed will effect where the ball will make contact with the bat. He may be late with the swing or quicker with swing so that will effect where the ball goes.