The Physics of Fungos

Ed Roebuck was not the kind of player who usually gets remembered. He pitched his way to a lifetime record of 52-31 from 1955 to 1966 playing for four different teams and gathered a grand total of -0.7 WAR. And yet people do remember him, not least because he somehow managed to earn World Series rings with the 1955 Dodgers and the 2004 Red Sox (for whom he was a scout). Roebuck’s SABR Biography is an interesting read, but it’s one particular aspect of his baseball career that interests us today. He was one of the very best hitters of fungos ever. From the aforementioned SABR Biography:

…One of his hobbies as a child was hitting stones with a stick or a club by himself. This turned out to be great training for such feats as hitting the eagle atop the scoreboard at the first Busch Stadium, hitting a ball over the roof at Forbes Field, hitting the center-field wall at the Polo Grounds, and whacking a ball into the colonnaded end of the Los Angeles Coliseum. For that feat, [Dodgers manager Walter] Alston fined Roebuck $75.

‘There is a funny story regarding how I got that $75 back. The architect designing the Astrodome asked Walter O’Malley how to figure how high to make the roof. O’Malley promised to find out how high his good fungo hitter, which was me, could hit one.

On the breakfast line at Vero, O’Malley approached me and asked me how high I could hit a fungo. I said, “I guess about 200 feet in the air, sir.” At the end of the workouts that day, Mr. O’Malley had me report to Field 4, where I hit a bunch of fungoes until he was satisfied. When I finished he asked me, “How much did Alston fine you for hitting the ball out of the Coliseum?” I told him $75, and a few weeks later a batboy came to the bullpen in spring training with a bag filled with $75 in quarters, which he said were from Mr. O’Malley.’”

Maybe you don’t know what the heck a fungo is. It is a bat designed specifically to hit balls tossed upward by a hitter as opposed to balls thrown by a pitcher. For example, a coach might use a fungo to hit balls for infield practice. A sample fungo is shown below. Why would a fungo be any different than a bat used by a player in a game? That’s where the physics comes in.

Think about the Home Run Derby. In about two minutes, a slugger takes roughly 25 swings. At that point, he almost always calls “time out” because he is exhausted. Now, imagine the situation of some poor coach who needs to hit fly balls to his outfielders for half an hour. A fungo bat is designed to address that and make it easy to hit pop flies or grounders with minimal effort.

Let’s try to understand how this is accomplished. Perhaps you’ve never noticed, but it is far easier to swing a bat if you hold it on the barrel as opposed to grabbing the handle. Physics tells us that it is always easier to get an object rotating if more of its mass is closer to the center of rotation. Why then do players insist upon using the handle?

You know the answer: if batters swung using the barrel, pitchers would routinely snap off the handle of the bat. Also, when you hit something with a tool like a hammer, you want as much of the mass as possible “on the business end” as it were.

Since a fungo bat isn’t designed to hit a pitched ball, it can be much thinner than a regular bat. After all, a batter facing a pitcher must change the incoming 95 mph pitch into a ball leaving the bat with a 95-mph exit velocity. That total change in the velocity is 190 mph–95 mph to bring the ball to rest momentarily on the bat plus another 95 mph to speed it back up again the other direction. However, a coach hitting a ball after an upward toss needs only to change the velocity of the ball from zero to 95 mph.

Let’s get specific and consider two different bats. According to the Angels’ website, Mike Trout uses an MT 27 Old Hickory brand maple bat, which has a 2.5 inch barrel and a 29/32 inch handle. It is 33.5 inches long and weighs 32 ounces. Louisville Slugger sells a Maple Wood Fungo Bat (#S345) that weighs between 20 and 22 ounces and is 35 inches in length. The fungo is clearly longer, lighter, and generally thinner than the regular bat. Below is a photo of the two bats side by side.

The effort needed to swing a bat is often described in baseball by the term “swing weight.” The related concept in physics is “moment of inertia,” or MOI. The connection between these ideas is described here. To calculate the MOI, one needs the exact profile of each bat. I used the photo above to find the radius of each bat in pixels at 20-pixel intervals along each bat. Knowing the length of each bat allows the pixels on the screen to be converted to inches on the bat.

Using this data and the weight of each bat, I then calculated the MOI about the knob of the bats. The MOI for the fungo turns out to be about 30 percent smaller that it is for the Trout bat, meaning that it is roughly 30 percent easier to swing. Now, the only question is whether it can hit the ball as hard.

You might recall the magic equation that predicts the speed of a well-hit ball based upon the speed of the bat and the speed of the pitch. If you don’t keep the equation on a slip of paper in your wallet, here it is for future reference,

where q is called the “collision efficiency.” You can find the origin of this equation here.

The collision efficiency depends upon lots of things like the mass of the ball, the mass of the bat, the MOI, the location along the bat where the collision with the ball happens, and the bounciness of the ball (coefficient of restitution). Using some reasonable values for a well-hit ball off the Trout bat gives a q of about 0.2.

Assuming v_bat = 70 mph and v_pitch = 90 mph, the equation tells us that the ball will be going about 102 mph off Trout’s bat. However, using a fungo, v_pitch is zero, so the second term in the equation is gone,

Using the values for the fungo gives a q of only 0.12. If one assumes the coach can swing the fungo at the same speed as before, the exit speed of the ball is only 78 mph, too slow for game-like fielding practice.

However, we have forgotten the fungo is about 30 percent easier to swing, so perhaps even an older coach taking it easy could manage a bat speed 15 percent higher than he could with Trout’s bat. Instead of a bat speed of 70 mph, the bat would be going about 80 mph, giving a batted ball speed of 90 mph sufficient for fielding practice.

At the time of many of Ed Roebuck’s fungo feats, he was an active player. Perhaps he could have actually swung the bat 30 percent faster. If so, the speed of the ball would have been 102 mph, which might explain some of his accomplishments.

So, now not only do you know why a fungo is ideal for hitting grounders all afternoon, but you’ve learned how the first indoor ball park was designed. Or so the story goes.

References and Resources

• Paul Hirsch, SABR Bio Project, “Ed Roebuck“
• Dr. Alan Nathan, American Journal of Physics, Vol. 71, No. 2, February 2003, “Characterizing the performance of baseball bats“
• Dr. Daniel Russell, Physics and Acoustics of Baseball & Softball Bats, “Swing Weight of a Bat (Why moment-of-inertia matters more than weight)“