As I briefly laid out in The Science Of Baseball: Coefficients And Happy Haitians, people like home runs though baseball purists don't necessarily think much of them - unless their team gets one.

Example: I was at yesterday's Pirates-Phillies game and the wind was blowing out about 30 MPH toward left center.  When spindly Pirates center fielder Andrew McCutchen came up in the 5th inning and hit what would have been a long out in most Little League parks, my brother John, a Phillies fan, insisted I put an asterisk next to it on my scorecard (1) to represent the shamefulness of it all but when mighty Phillies first baseman Ryan Howard hit a home run, the wind was apparently not a factor.  In fact, my brother was prepared to make a case that the wind had actually been blowing in at that moment, making Howard's accomplishment in the face of such natural adversity that much more spectacular.

Home runs bring out the passion in baseball fans and are the only thing worth noting to non-fans.   Non-fans do not appreciate pitching duels while baseball fans regard home run festivals as evidence of shoddy baseball.

Regarding home runs, the one question I often get is, 'What is the longest home run ever hit?'   You'd think in a game like baseball, where almost everything is numerically quantifiable, that would be an easy one.  But it isn't; anecdotal evidence is not much help, since every baseball player when an old guy was young was supposedly better than every baseball player of today and measurements were not really taken before 27 years ago.  But we have some idea what home runs were like in the past and can use the awesome power of physics to sort truth from  sentimentality.

A few things will make the difference in how far a home run goes.   As I discussed last time, anomalies like Happy Haitians make the exact drag coefficients of an individual baseball from day to day, or even year to year, difficult to know, but we know it isn't like golf - there is a narrow range of 'liveliness' a ball can have.   We can expect a variance of perhaps 10% in drag coefficient which would translate to 14 feet for a 400 ft home run.

But drag also has another odd thing it causes - movement.   On a non-spinning ball, velocity basically has to overcome drag and that's it.   But most baseballs are spinning quite a bit and asymmetric spinning plus asymmetric stitches (meaning asymmetric drag) can also cause a ball to do funny things - like curve.

A transverse Magnus force, directed at right angles to the direction of the air velocity and to the axis of the baseball's spin, is what this is - and it is proportional to spin frequency, ball velocity and the drag coefficient.

Velocity      Drag

Physical tests being what they are (meaning that I can find just as many real world tests that agree as disagree on any set of balls) we instead have to rely on a model, which will be a reasonable approximation, to get an answer to how far a ball can be hit and therefore how far a particular ball was hit based on the conditions that day.    (2)

So given what we know about Magnus force now - that it will make a ball move more but reduces drag and leads to a longer distance if hit - we also need to think about the air, just a little.    The force on a ball will impact its velocity and air density will certainly accomplish that.    A 400 foot long out at Yankee Stadium, near sea level, will go over 430 feet in Denver, a mile in the air - and probably be a home run.

Back to business; what is the farthest home run?

Before there were measurements, '500 foot home runs' were commonly stated but that's because people did not understand trajectories; people assume they are even so its distance when it reaches its apex will be mirrored in distance on the far side.   But gravity and drag do not work that way.  Using some calculations you can also duplicate (2) you can see the trajectories drop a lot even for a ball hit at an optimum 35 degree angle.

Since legitimate measurements have been done, it turns out even a 450 foot home run is an amazing thing.  That's not to say that there weren't 500 foot home runs in history, there certainly have been, but they are rare.

The legends about Mickey Mantle are many - and he was a great player.  One of the most commonly told is that in 1953 he cleared the 55 foot high left-center-field bleachers at Clark Griffith Stadium in Washington, a distance of 460 feet  - something that had never been done before.    Estimated home run distance: 565 feet, the longest in history outside vague reporter claims of 600 foot homeruns by Babe Ruth on a few occasions.   The problem; a kid picked up the ball and that was the measurement they used.  Was it a monstrous home run?   Absolutely.   It isn't like anyone else, not in decades of Major League or Negro League competition, had ever cleared those bleachers, but 565 feet is in the grey area of impossibility (3).

Square is where it was claimed to land, circles are likely distance each second after impact, bleachers are bleachers.   The Physics of Baseball, Robert K. Adair, Harper Perennial, New York, 1994

How so?   As I laid out above, a baseball at an optimum angle of 35 degrees with a  velocity of 110 MPH would go about 750 feet in a vacuum but that's still only 400 feet at sea level in New York.   So 565 feet would surely the greatest home run of all time.   Did it happen?  Almost certainly not, but it would have to have gone over 500 feet - because the bleachers were 55 feet high and 460 feet away from home plate.  (4)

Spectators said the ball glanced off a beer sign 60 feet above playing field height.  If he had a following wind like in yesterday's Phillies-Pirates spring training game (5) , 20 MPH or so, and an angle  higher than optimum to account for that, 40-50 degrees, he could have hit it 510 feet.

Why not 565?   Even assuming a 95 MPH pitch, which also makes the ball velocity higher on the way out (and no one ever claimed Washington pitcher Chuck Stobbs was a hard pitcher, so even 95 may be generous) and high heat and humidity that would have made the ball livelier, it could still only boost his distance to perhaps 530 feet.

But as I say in note 3 above, the Mick never hit in Denver.   Add another 9% for high altitude to those conditions (though it's never humid in Denver) and he could have sent one 580.

Does Mantle have the record?   The farthest accurately measured home run I can find is Cecil Fielder's 502 foot blast over the left-field bleachers at Milwaukee's County Stadium.  Not to take anything away from the Dave Kingman's (one he hit/rolled to the third house beyond Waveland Avenue at 530 feet) or Frank Howard's of the world but Mick probably beat them.  However, we can't count out Babe Ruth; at a time when there were a dozen reporters covering every game he played in and there were dozens of reports of 600 foot home runs, the idea that he hit many 500 foot homeruns is very plausible; even more impressive is that he also likely did it in all of the stadiums the American League played - which means he would have hit at least one farther than 510 feet.

NOTES:

(1)  About the scorecard.   I had asked my our trustworthy communications liason, Kristina, to get me the 'game day' media kit from the Pirates because I thought I might do a pitch-by-pitch analysis and the game day handouts will have up-to-the-minute stats, including match-ups for various combinations and historical stuff.  It's the kind of thing the color commentators use when they are talking about players during the game.

The Pirates media relations guy, Jim Trdinich, told her it was too late on Wednesday to get the stats on Saturday but I could download some from the web.    Now, you'd think a media relations guy representing a team that is tied for the most prolonged period of incompetence in major sports history (16 consecutive losing seasons, fittingly the Phillies  of 1933-48 are the other team in the same boat) and will maybe own it all to themselves this season, would be falling over backwards to get some positive press, since we have 4X the readership per month of the Pittsburgh Post-Gazette but, no, three days advance notice was not enough.

Instead, my brother and I scalped some fine seats at 50% off just above first base and I made do with the score card in the \$5 program.    This scorecard design was ridiculous, I can't scan it in here or I would, but basically it is supposed to look like a spiral notebook piece of paper, except the strange graphical marks to simulate a binder are in the actual scoring section and they are light grey - the same color as the pencil marks I was making.

So I sat there complaining about my scoring looking like heiroglyphics due to the odd graphical design of the thing ( seriously, who does that to a scorecard?)  when a kindly grandmother in front of me with a stack of papers turned to me and said, "Would you like this score card?  I heard you were having trouble with yours and I got this from a friend who is a reporter but I won't use it."   Of course, she is holding the media kit that apparently needed to be kept under lock and key...

(2) REYNOLDS NUMBER - A sphere of diameter r with a velocity V through a fluid of density ρ and viscosity μ is:   R= ρVr/μ
DRAG FORCE - V in MPH.   A=πr2.  ρ is air density.  Cd is drag coefficient.    Fd =1/2CdρAV2

(3) Mickey Mantle did not have the chance to hit in Denver.

(4) The Mick  was a humble man but he gives us an idea of his power when he was asked about what he thought was his greatest home run.    He did not say that one in Washington but rather one he hit in May of 1963 at Yankee Stadium.

Mick's home run hit the facade on the right-field roof  about 370 feet from home plate; that's 115 feet above field level - and it was still going up, people said, meaning it could have gone 620 feet, according to math.  First, that is just not possible.   If even Mickey Mantle had the strength to have a ball still going up when it hit that facade,  it would have cleared it by 100 feet; it just doesn't work any other way, miracles of eyesight notwithstanding.  The angle was all wrong though if he actually had the strength to have it still rising at 370 feet, and hit at at the optimum 35 degree angle, he would have cleared the facade by 200 feet; yet still not gone 620 overall.

(5) Dan Valenti, in Clout, quotes Sam Diaz, the meteorologist working that day in the Washington bureau, who says the wind was 20MPH with gusts up to 41MPH.

REFERENCES:

Clout, Dan Valenti, Stephen Greene, New York, 1989

The Physics of Baseball, Robert K. Adair, Harper Perennial, New York, 1994