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Measuring Home Run Distance
On May 22, 1963, Mickey Mantle, the great New York Yankee slugger, hit a prodigious home run; some say the longest ball ever hit in the major leagues. Although it struck the stadium roof only 370 feet from home plate, some news reports credited Mantle with a 620-foot blast. Why was there this vast discrepancy? By tradition, when a batter hits a home run and it lands somewhere within the ball park, the distance this home run is said to have traveled is not the distance to the point of impact, but the estimated distance the ball would have traveled had its flight been unobstructed. As we shall see, in Mantle's case, this distance may have been considerably exaggerated.
Illustration of trajectory
Four parameters come in to play in estimating home run distance:
1. The ball’s height above the field at the point of impact with the stadium ( h in the diagram).
2. The horizontal distance that the ball has traveled when it impacts the stadium ( s in the diagram).
3. The angle of impact — the angle that the tangent line to the flight path makes with the horizontal 
        ( A in the diagram).
4. The height at which the bat struck the ball (k in the diagram).
To calculate the distance, we view the parameters h, s, A, and k as given and assume that the path of the batted ball is a parabola, at least from the point of impact with the stadium to the “virtual impact point” with the ground — the spot at which the ball would have hit the ground had its path been unobstructed. Traditionally, these parameters have been estimated with the aid of a tape measure and careful observation of the ball’s flight path. Today, sophisticated technology (see, for example, reference [4]) can provide much more accurate figures for these variables.
We then introduce a Cartesian coordinate system in the plane of the ball’s path, in which the origin is at home plate and the positive x-axis runs along the field in the direction of flight (see the diagram above). In this coordinate system, we assign the coordinates (d, 0) to the virtual impact point, which makes d the “home run distance.” To determine the equation of the desired parabola, y = ax2 + bx + c , from the known parameters h, k, s, and A, we make use of the following conditions:
Baseball animation
Note:  Both trajectories have 
the same "baseline" distance.

The Calculation

(1) When x = 0, y = k.
(2) When x = s, y = h

(3) When x = s, dy/dx = tanA

Condition (1) yields c = k. Conditions (2) and (3) result in the linear system (with unknowns a and b):
h = as2 + bs + k

tan(A) = 2 as + b

Solving this system, we see that 

a = [ s(tanA) - h + k] / s2  and  b = [2h - s(tanA) - 2 k] / s .

Once a and b have been calculated, the home run distance, d , is determined by finding the positive solution of the quadratic equation ax2 + bx + k = 0..

As an example, suppose a batter hits a home run into the seats. A chart supplied by the home club tells us that the ball landed 50 feet above the playing field and 400 feet from home plate. Thus, h = 50and s = 400.We estimate that the bat met the ball 3 feet (k) above home plate and that the angle of impact with the stadium was 135 E ( A). Then, using the formulas given above, we see that the path of the ball has the equation y = ax2 + bx + c, where a = -0.00279, b = 1.235, and c = 3. So, the home run distance, d, is the positive solution of the equation 

(-0.00279)x2 + (1.235)x + 3 = 0.

Using the quadratic formula, we see that d is approximately 445 feet. (Note that the actual (straight line) distance that the ball traveled to the point of impact, obtained from s and h by using the Pythagorean Theorem, is approximately 403 feet.)

Note:  Please see references [2] and [5], p. 594, for other approaches to this problem.
Let’s now return to the Mantle home run that was mentioned at the top of this page. Its point of impact with the stadium was approximately 115 feet above field level (h in the formula) and 370 feet from home plate (s ). The 620-foot estimate for this blast was based on the belief that the ball was near the peak of its travel at the point of impact with the stadium, which has since been deemed to be highly unlikely (see references [1], pp. 104 - 105 and [3]). Suppose we take the more conservative view that the path of the ball made an angle of 150 E ( A in the formula) with the horizontal when it struck the stadium. We will also assume, for the sake of simplicity, that Mantle “golfed” the pitch off the ground, so that k can be taken to be 0. (This simplification alters the computed home run distance by less than a foot.) From this data, we compute the parabolic path of the ball to be y = ax 2 + bx,where a = -0.00239and b = 1.199.So, the home run distance, d, is the positive solution of the equation
(-0.00239)x2 + (1.199) x = 0,

which is approximately 502 feet.

The baseball

Baseball cover

is formed by folding two identical perpendicular plane regions, or "curves," to form a sphere.  Its circumference is 9 1/8 inches, a length that permits a pitcher to get a firm grip on the seams.

The cover was invented by a young boy, Elias Drake, in the 1840s.

[1]  W. Adair, Robert,  The Physics of Baseball (3rd ed.), Harper and Collins, 2002.

[2]  HowStuffWorks, Question of the Day,

[3]  W. Jenkinson, Long Distance Home Runs, 1996.

[4]  Sportvision, HR Distance, 2002.

[5]  Varberg, Pursell, and Rigdon, Calculus (8th ed.), Prentice Hall, 2000.

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Stewart Venit
Summer, 2002