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I am not much of a sports fan and when the newspaper comes each morning, I usually toss the sports section in the recycling bin. But, when I am working a wind tunnel test out of town, there is usually quite a bit of dead time where I end up reading everything from recipes to lost dog notices and even sports columnists. So, it happened one day that I was perusing The Dallas Morning News and I came across the assertion that a baseball will travel nine percent further in the new Denver stadium than at sea level. If the writer had just said that it would travel further, I probably would have mumbled something about the air density being lower and passed on. But, when it said that it will travel nine percent further - not eight percent, not ten percent, but nine percent further - I had to know just where that number came from. Of course, my interest was also professional, because the essential thing you will need if you are going to make calculations of the air resistance is a procedure for computing the density of air at any altitude. And you have certainly come to the right web site for that because you can jump to my Standard Atmosphere page and read about it.

Baseball Trajectory Program

Almost all elementary textbooks on physics or mechanics show the procedure for calculating the trajectory of an object that is given a certain initial velocity. The solution is always done with the assumption that air resistance is negligible and gravity is the only force acting on the object. This assumption makes it possible to solve the equations and produce a solution that lets you compute the position and velocity of the object at each point along the path. You can compute the horizontal distance travelled, the maximum height reached halfway along the path, and even show that you get the best range if you throw the object at an angle of 45 degrees.

How do we compute the trajectory if we do not neglect air resistance? As we learn in elementary mechanics, we identify all of the forces acting on the object, write F=ma for both the horizontal and vertical directions and then proceed to solve the resulting set of differential equations. Writing the equations is not too hard, but the solution is not easy because the aerodynamic drag force varies with velocity. The direction of the drag force changes as the object travels along its path, always being opposite in direction to the velocity. If you want to go through the details of the dynamical equations and the numerical solution of the differential equations, go to the Numerical Procedure page.

Results of the Calculations

To perform the calculations, we must specify the initial conditions (altitude and velocity) of the ball and then follow the trajectory until the altitude returns to the initial value. If we do this for Denver with an altitude of 1609 m, and several possible initial velocities, we get the following results:

Initial
Velocity
(m/s)
Sea Level
Range
(meters)
Denver
Range
(meters)
% gain
35 72.9 77.2 6.0
40 85.3 91.0 6.8
45 96.9 104.2 7.5
50 107.7 116.5 8.1
55 117.8 128.1 8.7

It turns out that the ball does indeed travel further in the less dense air of Denver. It also turns out that the percent improvement varies with initial velocity. The high velocity balls that will travel a long distance will gain more range than the slower ones. So, I guess I will go along with the News and say that the guys who hit really long balls can get a nine percent improvement in distance travelled. For us wimps who have a hard time getting 30 m/s, the benefits are significantly smaller.

Speculations about other ballparks

If we get longer fly balls in Denver, why not build stadiums in Mexico City or even La Paz, Bolivia? For our ultimate fantasy, lets imagine a stadium near the summit of Mount Everest and another one in which all the air has been removed and the game is played by androids or humans with scuba packs. If we repeat the calculations, we get the following results:

Initial
Velocity
(m/s)
Sea Level
Range
(meters)
Denver
Range
(meters)
Mt. Everest
Range
(meters)
Vacuum
Range
(meters)
35 72.9 77.2 94.5 123.1
40 85.3 91.0 115.2 160.8
45 96.9 104.2 135.7 203.5
50 107.7 116.5 155.8 251.2
55 117.8 128.1 175.4 304.0

Now we are getting some really long balls. Our 55 m/s ball that would travel 118 meters at sea level will now go 304 meters in the vacuum ball park. WOW! Two and a half times as far. That shows that air resistance is quite important.

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