Projectile motion: Most projectile motion problems are derived in a plane (2-dimensions only). But we live in a three-dimensional world, and objects do not always move in flat planes. (Consider skier coming down a hill, swooshing back and forth in an S-pattern while descending the hill.) This problem will use the projectile motion formulas and extend them to a three-dimensional setting. Suppose a (level) college baseball field is oriented so that the first base foul line (which we will take t be the positive x-axis) goes straight east from home plate, and the third base foul line (= positive y-axi goes straight north from home plate. This puts "dead center field" at exactly NE of home plate (at a angle to the foul lines). Put the origin (0, 0, 0) at home plate. All distance scales (down the foul lines and vertically) are in fe Time t will be measured in seconds. Our formula for the projectile motion of a baseball hit exactly above home plate by a batter will denoted r(t), and will have three components < x(t), y(t), z(t) >. To derive the standard path of the baseball (as a parabolic projectile motion), one would start with the acceleration function a(t) = < 0, 0, -g >, where g is the acceleration of gravity constant. Because of choice of dimensional units, we will use g = 16.1 ft/sec. Integrating twice (as you did in #3 above) will produce your position function. (I believe this is done in the book.) A baseball is hit with an initial speed of 120 ft/sec at an angle of elevation of 0 = 250 at an initial height of exactly 3 feet above home plate. This means that v(0) = < (120 cos 25") cos a, (120 cos 25") sin a, 120 sin 25">, where a = the ang measured from the positive x-axis (= first base foul line) giving the initial xy-direction of the baseball. For example, we will consider three directions for the initial xy-direction of the baseball: a = 0' is down the positive x-axis (= first base line), a = 90" is down the positive y-axis (= third base line), and a = 45" is to "dead center field" (= across the pitcher's mound to the points where x = y) blem #5 (continued) e graph at right shows the baseball field. ere is a circular 12-ft. tall fence located distance of 350 feet from home plate ween the foul lines. seballs hit over this fence are home runs! dotted line extends from home plate to ad center field" on a 45 angle from h foul line. this graph to solve the problems below. intain at least five decimal places (or re) for all intermediate steps; round final wers to the nearest 0.1 ft and 0.1 sec. State the position function r(t) for the Third base trajectory of the baseball in terms of the angle a. Leave your formula in exact symbolic form (without rounded decimal values). First base Home plate Find the maximum height for the baseball's trajectory and the time at which it occurs. This answer does not depend on the value of a, so choose any convenient value of a for this question. Find the (x, y, z)-coordinates of the point and the distance from home plate where the baseball lands or the ground for each of the following values of a: (i) a = 0 (ii) a = 450 (iii) a = 90 nich (if any) of these baseball trajectories were a home run (meaning it landed beyond the 350-foot fenc ND they cleared the 12-ft height at the 350 foot mark? Now let's introduce a realistic additional factor for this problem: wind! Assume there is a steady 9 mile per hour wind (= 13.2 fu/sec) blowing in a NE direction that is 30" from the positive x-axis (= first base foul line). This wind vector = w = < (13.2 cos 30")t , (13.2 sin 30")t is creates a new trajectory: For each a-value the new position function = R(t) = r(t) + w(t). Redo problem (c) above for the same three values of a with the new R(t) position function (with win If the baseball clears the fence, give the (x, y, z)-coordinates where the baseball hits the ground (in the zone past the fence). [Assume the ground past the fence is on the same level as the field - there are n outfield stands like T-Mobile Park.) If the baseball doesn't even make it 350 feet away from home plate, give the (x, y, z)-coordinates of where the baseball hits the ground (in the field of play). If the baseball hits the fence, determine the (x, y, z)-coordinates where it hits the fence. minder: Keep at least five decimal places or more for intermediate calculations, but round final swers to the nearest 0. 1 feet for each coordinate