Measuring Home Run Distance
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
credited Mantle with a 620-foot blast. Why was there this vast
By tradition, when a batter hits a home run and it lands somewhere
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
have traveled had its flight been unobstructed. As we shall see, in
Mantle's case, this distance may have been considerably exaggerated.
parameters come in to play in estimating home run distance:
height above the field at the point of impact with the stadium ( h
in the diagram).
distance that the ball has traveled when it impacts the stadium ( s
in the diagram).
of impact — the angle that the tangent line to the flight path makes
( A in the diagram).
at which the bat struck the ball (k in the diagram).
the distance, we view the parameters h, s, A, and k as
and assume that the path of the batted ball is a parabola, at least
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
path been unobstructed. Traditionally, these parameters have been
with the aid of a tape measure and careful observation of the ball’s
path. Today, sophisticated technology (see, for example, reference )
can provide much more accurate figures for these variables.
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
the equation of the desired parabola, y = ax2
+ bx + c , from the known parameters h, k, s,
A, we make use of the following conditions:
Note: Both trajectories have
the same "baseline" distance.
(1) When x = 0, y
(2) When x
(3) When x = s, dy/dx = tanA.
Condition (1) yields c
Conditions (2) and (3) result in the linear system (with unknowns a
h = as2
tan(A) = 2 as + b
Solving this system, we see
a = [
- h + k] / s2 and b = [2h
- s(tanA) - 2 k] / s .
have been calculated, the home run distance, d , is determined
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
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
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
+ (1.235)x +
3 = 0.
formula, we see that d is approximately 445 feet. (Note that
actual (straight line) distance that the ball traveled to the point of
impact, obtained from s and h by using the Pythagorean
is approximately 403 feet.)
Note: Please see references
 and , p.
594, for other approaches to this problem.
x = 0,
Let’s now return to the
run that was mentioned at the top of this page. Its point of impact
the stadium was approximately 115 feet above field level (h in
formula) and 370 feet from home plate (s ). The 620-foot
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 , pp. 104 - 105
). Suppose we take the more conservative view that the path of the
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
“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
a foot.) From this data, we compute the parabolic path of the ball to be
y = ax 2 + bx,where a =
b = 1.199.So, the home run distance, d, is the positive
of the equation
is approximately 502 feet.
is formed by folding two identical
plane regions, or "curves," to form a sphere. Its circumference
9 1/8 inches, a length that permits a pitcher to get a firm grip on the
The cover was invented by a young
Drake, in the 1840s.
W. Adair, Robert, The Physics of Baseball (3rd
ed.), Harper and Collins, 2002.
HowStuffWorks, Question of the Day,
Jenkinson, Long Distance Home Runs, 1996.
Sportvision, HR Distance, 2002.
Pursell, and Rigdon, Calculus (8th ed.), Prentice