Page 265 - Excel for Scientists and Engineers: Numerical Methods

242 EXCEL NUMERICAL METHODS

velocity, f = Dv2. The proportionality constant D = 0.5pCA, where p is the
density of air, A is the cross-sectional area of the projectile and C, the drag
coefficient, is a dimensionless quantity that depends on the shape of the
projectile. The forces acting on a projectile in flight are illustrated in the
following figure.

g
Combining the above equation for the air drag and the relationship between
force and acceleration, f = ma, we get, for the "deceleration" in the x-
direction, x" = -Dv:/m; y" = -Dv;lm-g.
Calculate the trajectory of a baseball hit at angle 8= 30" with initial velocity
50 m/s. The parameters of the baseball are: mass 145 g, circumference 23
cm (from Rules of Baseball, Major League Baseball Enterprises, 1998). For
air resistance, use p = 1.2 kg/m2 and the drag coefficient C = 0.5.
y", XI, y', v, x, y. At t
(Getting started: create eight columns, as follows: t, XI',
= 0, XI and y' are calculated as in the previous problem, but for subsequent t
values, they are calculated by the Euler method, using the previous values of
XI' and y". Calculate x and y, the coordinates of distance traveled, using, e.g.,
xt+l = xt + x;At + '/2Xl"(At)2.)

7. Pendulum Motion I. The motion of a simple pendulum, consisting of a
mass Mat the end of a rod of length L, is described by the following first-
order differential equation:
dw -g sin8
-=--
d8 L w
where w = angular velocity (radls)
8= angle of displacement from equilibrium position
g= 9.81 m/s2
L= 1.0m
Calculate the angular velocity of the pendulum beginning with the initial
conditions 8= lo", w= 0.3.