3. Baseball dynamics depends on the ball's seam, the air drag, the initial rotational and linear speeds and the orientation of the ball. Consider a left-handed pitcher pitching a 'slider' to right-handed batter (see attached figure). Assuming that x measures displacement from the pitcher to the batter, z measures vertical displacement of the ball (+z is upward displacement direction (against gravity)), and y measures horizontal displacement (+y corresponds to the displacement to the hitter's right-hand side), vo is the initial speed of the ball, is the angle made by the ball's trajectory with the x axis (see figure), is the angular velocity vector written as = w(0, sin(), cos(0)), where is the angle made by the angular velocity vector with the vertical axis (2). For a slider, = 0. Let the distance between the pitcher and the batter be 18.44 m. The initial rotational spin is w| = 1800 rpm with = 0°. Since CC is a left handed pitcher, the spin will be in the reverse direction, and so use w = -1800 rpm. The trajectory and the speed of the baseball is governed by the following system of equations: dt dy = dx = Ux |||L| Vy dt dz dt dvi dt = = Vz -F(v)vvr + Bw(v2sin(o) - vycos(0)) (5) 4500 (4) (6) (7) N 13 4feet slider 18.44 m Pitching mound Batter = -F(v)vvy + Bwvzcos(ó) (8) dv. =-g-F(v)vv, + Bwv,sin(o) dt (9) where v₂, ty, and v₂ are the velocity components in the r, y, and z directions, respectively. Also, v corresponds to the speed of the ball v = √√+v+vg = 9.81 is the gravitational acceleration, and F(v) is the drag force experienced by the baseball which is given as, 0.0058 F(x)=0.0039 + 1+ exp[(v-35)/5] (10) in SI units. B in the above expression results from a force called Magnus Force, a force due to rotation of the ball in a shearing flow that causes the ball to curve during its forward flight. Use B = 4.1 x 104 (note that B is dimensionless quantity). Let the following be the initial conditions for the above system of equations: x(t = 0) = 0 y(t = 0) = 0 z(t = 0) = 0 vy(t = 0) = 0 v(t = 0) = cos(6) v₂(t = 0) = vosin (0), (11) (12) (13) (14) (15) (16) where v = 85 mph is the initial speed, and 0 = 1°. (a) Using Forward Euler discretization scheme, obtain a finite difference approximation for each of the equations. Because of the non-linear equations, it is not necessary and not recommended to write this in matrix-vector form. (b) Solve above system of differential equations (i.e. write a program that will solve the above equations using Forward Euler) and obtain solution up to a time at which the ball crosses the batter (i.e. travels a total distance of 18.44 m. Be careful with the unit conversions (convert all quantities to SI units). What is the time required for the ball to cross the batter in seconds? What is the speed of the ball (in miles per hour) when it just crosses the batter? Choose At = 1 x 10 s. (c) Plot z (i.e. vertical displacement of the ball) against r (r on x-axis). On the same plot (and using different line style) plot y (i.e. horizontal displacement of the ball) against 1. Clearly label your plots. [For plotting purposes, convert r, y and z in feet rather than meters.] What is the net vertical displacement of the ball (note positive displacement is vertically upwards) as it crosses the batter? What is the net horizontal displacement of the ball as it crosses the batter?