A. Determine the density of air at Mile High Stadium in Denver, Colorado, on a w

Question

A. Determine the density of air at Mile High Stadium in Denver, Colorado, on a warm summer evening when the temperature is 78 F. The barometric pressure is 85.1 kPa.

B. Assume to a first approximation that the drag froce exerted on a well-hit baseball arcing to the outfield, or beyond, is proportional to the air density. Discuss the implications of this for balls hit at Mile High Stadium in Denver versus balls hit in Yankee Stadium in New York.

C. Find the approximate ratio between the distance a ball will travel when hit with the same force in New York as compared to Colorado.

Explanation / Answer

a) Using Ideal gas law we can introduce air density as a function of pressure and temperature

air = P/R.T where P= pressure= 85.1KPa = 85.1 *10^3 Pa

R= specific gas constant , (we consider dry air as the evening is warm)= 287.05J/Kg.K

T= temperature= 78F = 298.7K

using them we get, air = 85.1*10^3 / 287.05 * 298.7 = .9925 Kg/m^3

b) According to the question baseball drag force is proportional to the air density

that means F air

Using other variables like drag coefficient=C, velocity of air =V and reference area =A

the expression becomes,

F= Cair AV^2 Now standard drag coefficient of a base ball using Reynolds number is .3

therefore F= .3air AV^2

standard model includes a value .5 in it

therefore F= .15air AV^2

Standard radius of a baseball is .12ft= .03576m

Therefore A= r^2 = 3.14* .03576^2 = .004017m^2

Now Denver is in a height of 5280ft and standard air density is 82% of sea level density

again Newyork 32.8ft above sea level and air density is 2% below average (1.29Kg/m^3)

V in new york = 13km/h = 3.6m/s

V in denver = 7.2m/s

Using them Drag force on Mile high F= .15*.9925* .004017* 7.2^2 =.031 N

Drag force on Yankee F= .15*1.29 *.004017*3.6^2 = .01007N

That means drag force in Denver is greater than Newyork

c) Using V^2 = U^2 +2gS where U=0 for both case

and V for Denver=7.2m/s and V for Newyork =3.6m/s

we get for Denver,

7.2^2 =0 + 2*9.8 *S or S= 2.64m

for Newyork

3.6^2 =0+ 2*9.8*S or S= .6612m

The ratio of covered distances Newyork : Denver = .6612/2.64 = .25 (approx)

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