Applications - A Guide to MATLAB for Beginners and Experienced Users - page 213

is the magnitude of the velocity, that is, the speed, and c is a constant
depending on the shape and mass of the ball and the density of the air.
(We are neglecting the lift force that comes from the ball's rotation, which
can also play a major role in some situations, for instance in analyzing
the path of a curve ball, as well as forces due to wind currents.) For a
baseball, the constant c turns out to be approximately 0.0017, assuming
distances are measured in feet and time is measured in seconds. (See,
for example, Chapter 18, “Balls and Strikes and Home Runs,” in Towing
Icebergs,FallingDominoes,andOtherAdventuresinAppliedMathematics ,
by Robert Banks, Princeton University Press, 1998.) Build a SIMULINK
model corresponding to Equation (7), and use it to study the trajectory of
a batted baseball. Here are a few hints. Represent x , x , and x as vector
signals, joined by two Integrator blocks. The quantity x , according to (7),
should be computed from a Sum block with two vector inputs. One should
be a Constant block withthe vector value [0 , − 32 . 2], representing gravity,
and the other should represent the drag term on the right of Equation
(7), computed from the value of x . You should be able to change one of the
parameters to study what happens both with and without air resistance
(the cases of c = 0 . 0017 and c = 0, respectively). Attachthe output to an
XY Graphblock, withthe parameters x-min = 0, y-min = 0, x-max = 500,
y-max = 150, so that you can see the path of the ball out to a distance of
500 feet from home plate and up to a height of 150 feet.
(a) Let x (0) = [0 , 4], x (0) = [80 , 80]. (This corresponds to the ball start-
ing at t = 0 from home plate, 4 feet off the ground, with the hori-
zontal and vertical components of its velocity bothequal to 80 ft/sec.
This corresponds to a speed off the bat of about 77 mph, which is not
unrealistic.) How far (approximately — you can read this off your
XY Graph output) will the ball travel before it hits the ground, both
with and without air resistance? About how long will it take the ball
to hit the ground, and how fast will the ball be traveling at that time
(again, both with and without air resistance)? (The last parts of the
question are relevant for outfielders.)
(b) Suppose a game is played in Denver, Colorado, where because of
thinning of the atmosphere due to the high altitude, c is only 0.0014.
How far will the ball travel now (given the same initial velocity as
in (a))?
(c) (This is not a MATLAB problem.) Estimate from a comparison of
your answers to (a) and (b) what effect altitude might have on the
team batting average of the Colorado Rockies.