Question 7.7.4: Batted baseball Suppose that a batted ball starts at x0 = 0,...
Batted baseball Suppose that a batted ball starts at x_{0} = 0, y_{0} = 0 with initial velocity v_{0} = 160 ft/s and with initial angle of inclination θ = 30°. If air resistance is ignored, we find by the elementary methods of Section 1.2 that the baseball travels a [horizontal] distance of 400 \sqrt{3} ft (approximately 693 ft) in 5 s before striking the ground. Now suppose that in addition to a downward gravitational acceleration (g = 32 ft/s²), the baseball experiences an acceleration due to air resistance of (0.0025)v² feet per second per second, directed opposite to its instantaneous direction of motion. Determine how far the baseball will travel horizontally under these conditions.
According to Problem 30 of Section 7.1, the equations of motion of the baseball are
\frac{d^{2}}{dt^{2}} = -cv \frac{dx}{dt} , \frac{d^{2}y}{dt^{2}} = -cv \frac{dy}{dt} -g (23)
where v = \sqrt{(x^{′})^{2} + (y^{′})^{2}} is the speed of the ball, and where c = 0.0025 and g = 32 in fps units. We convert to a first-order system as in (22) and thereby obtain the system
x^{′}_{1} = x_{3},
x^{′}_{2} = x_{4},
x^{′}_{3} = -c x_{3} \sqrt{x^{2}_{3} + x^{2}_{4}},
x^{′}_{4} = -c x_{4} \sqrt{x^{2}_{3} + x^{2}_{4}} – g (24)
of four first-order differential equations with
x_{1}(0) = x_{2}(0) = 0.
x_{3}(0) = 80 \sqrt{3}, x_{4}(0) = 80. (25)
Note that x_{3}(t) and x_{4}(t) are simply the x- and y-components of the baseball’s velocity vector, so v = \sqrt{x^{2}_{3} + x^{2}_{4}}. We proceed to apply the Runge–Kutta method to investigate the motion of the batted baseball described by the initial value problem in (24) and (25), first taking c = 0 to ignore air resistance and then using c = 0.0025 to take air resistance into
account.
WITHOUT AIR RESISTANCE: Figure 7.7.4 shows the numerical results obtained when a Runge–Kutta program such as rkn is applied with step size h = 0.1 and with c = 0 (no air resistance). For convenience in interpreting the results, the printed output at each selected step consists of the horizontal and vertical coordinates x and y of the baseball, its velocity v, and the angle of inclination α of its velocity vector (in degrees measured from the horizontal). These results agree with the exact solution when c = 0. The ball travels a horizontal distance of 400\sqrt{3} ≈ 692.82 ft in exactly 5 s, having reached a maximum height of 100 ft after 2.5 s. Note also that the ball strikes the ground at the same angle and with the same speed as its initial angle and speed.
WITH AIR RESISTANCE: Figure 7.7.5 shows the results obtained with the fairly realistic value of c = 0.0025 for the air resistance for a batted baseball. To within a hundredth of a foot in either direction, the same results are obtained with step sizes h = 0.05 and h = 0.025. We now see that with air resistance the ball travels a distance well under 400 ft in just over 4 s. The more refined data in Fig. 7.7.6 show that the ball travels horizontally only about 340 ft and that its maximum height is only about 66 ft. As illustrated in Fig. 7.7.7, air resistance has converted a massive home run into a routine fly ball (if hit straightaway to center field). Note also that when the ball strikes the ground, it has slightly under half its initial speed (only about 79 ft/s) and is falling at a steeper angle (about 46°). Every baseball fan has observed empirically these aspects of the trajectory of a fly ball.
