A Mathematical Analysis of Randy Johnson's bird killing fastball

[Hambil]

Hi, I'm new here. I thought this might be fun to discuss.

http://video.google.com/videoplay?docid=7763715434502216539&q=Randy+Johnson"

I need help on this one. We've got to consider the speed of the ball, which should be on record somewhere, or we can estimate it based on Randy's fastball speed. The speed of the bird can be guessed at via watching the video and comparing it with the known speed of the ball.

Then we have the number of birds in the stadium, how often (given the stadium volumn and number of birds) a bird is likely to cross through that specific area. And many other factors I'm sure I'm not thinking about.

I want to know just how rare a moment this was caught on tape. I suspect it is extrodinarily rare, like cosmicly rare.

 

[Quaoar]

Well, this is basically a mean free path question.

Take a look at the "Refinement of Mean Free Path" equation on this page:
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html

The only difference is that for the volume of targets swept, we will use the average velocity of a bird (We'll assume that birds and baseballs have roughly the same cross section). This means that the final equation will look like this:

v_fastball / (root(2) * pi * radius_of_baseball^2 * v_bird * bird_number_density)

If we plug in the numbers:
v_fastball ~ 40 m/s
v_bird ~ 10 m/s
radius_of_baseball ~ 5 cm
bird_number_density ~ 10 birds in a 100m by 100m by 100m volume (10e6 m^3) = 10e-5 birds per m^3 (This is a complete estimation, based on the number of birds I've seen flying around a baseball stadium).

This gives us a MFP of about 3.6e7 meters (or 36,000 kilometers).

Now, in each game, about 100 pitches are thrown by each team. Each pitch travels 60 feet, or 18 meters. So it would take 3.6e7 / 18 = 2 million pitches to hit one bird. In MLB, there have been around 25 teams, each of which has played around 150 games per year since baseball started being played professional, 100 years ago. That means there have been a total 150 * 100 * 100 = 1.5 million pitches thrown since baseball began.

Therefore, a bird getting hit by a baseball is totally plausible.

Keep in mind that these are all estimations, and for some things (such as bird density), I might be up to an order of magnitude off. But even if I'm off by a factor of 100, there's still a 1% chance a bird would have been hit by a pitch in the past 100 years. The number of opportunities increases if we include minor league games, amateur leagues, and games played in other countries.

I hope this answers your question... :)

 

[Hambil]

This also assumes an even distribution of birds/time around the entire stadium. But that wouldn't be true. They would spend much more time in some areas than others, since they are probably there for the cast off food.

Also, doesn't the MFP need to consider how often the bird and ball are in motion? The bird would be in motion (flying around the stadium) far more often then the pitches. A single pitch lasts maybe a second, so in an averege game the ball is in the air (counting pitches only) for only 100 seconds. While the bird may be flying around for half the game.

 

[Quaoar]

This also assumes an even distribution of birds/time around the entire stadium. But that wouldn't be true. They would spend much more time in some areas than others, since they are probably there for the cast off food.

Also, doesn't the MFP need to consider how often the bird and ball are in motion? The bird would be in motion (flying around the stadium) far more often then the pitches. A single pitch lasts maybe a second, so in an averege game the ball is in the air (counting pitches only) for only 100 seconds. While the bird may be flying around for half the game.

If you look at the link I provided, you can see that time is both in the numerator and in the denominator, so it cancels out.

Also, I assumed that the birds are constantly moving around at 10 m/s, which is roughly the flying speed of a bird. And, since I think quantifying bird density to a higher order is very park/environment-dependent, we have to use a more rough estimation. Like I said, my bird density figure might be a little high, but even if you reduce it by a significant factor, there's still a pretty significant chance it will happen in a Major League Baseball game.

Basically I converted the original problem in this: How far does a baseball have to travel before it hits one bird? Since we know how far a baseball travels per pitch, we can determine the number of pitches this is equivalent to.