855 8.8 Parametric Equations, Graphs, and Applications x y u Applications of Parametric Equations Parametric equations are used to simulate motion. If an object is thrown with a velocity of v feet per second at an angle u with the horizontal, then its flight can be modeled by x = 1v cos U2t and y = 1v sin U2t −16 t2 +h, where t is in seconds and h is the object’s initial height in feet above the ground. Here, x gives the horizontal position information and y gives the vertical position information. The term -16t2 occurs because gravity is pulling downward. See Figure 80. These equations ignore air resistance. y x v sin U v cos U v U Figure 80 EXAMPLE 6 Simulating Motion with Parametric Equations Three golf balls are hit simultaneously into the air at 132 ft per sec (90 mph) at angles of 30°, 50°, and 70° with the horizontal. (a) Assuming the ground is level, determine graphically which ball travels the greatest distance. Estimate this distance. (b) Which ball reaches the greatest height? Estimate this height. SOLUTION (a) Use the following parametric equations to model the flight of the golf balls. x = 1v cos u2t and y = 1v sin u2t - 16t2 + h Write three sets of parametric equations. x1 = 1132 cos 30°2t, y1 = 1132 sin 30°2t - 16t 2 x2 = 1132 cos 50°2t, y2 = 1132 sin 50°2t - 16t 2 x3 = 1132 cos 70°2t, y3 = 1132 sin 70°2t - 16t 2 The graphs of the three sets of parametric equations are shown in Figure 81(a), where 0 … t … 3. From the graph in Figure 81(b), where 0 … t … 9, we see that the ball hit at 50° travels the greatest distance. Using the tracing feature of the TI-84 Plus calculator, we find that this distance is about 540 ft. (b) Again, use the tracing feature to find that the ball hit at 70° reaches the greatest height, about 240 ft. S Now Try Exercise 47. Substitute h = 0, v = 132 ft per sec, and u = 30°, 50°, and 70°. 0 0 400 600 x3, y3 x2, y2 x1, y1 (a) 0 0 400 600 x3, y3 x2, y2 x1, y1 (b) Figure 81 A TI-84 Plus calculator allows us to view the graphing of more than one equation either sequentially or simultaneously. By choosing the latter, we can view the three golf balls in Figure 81 in flight at the same time. 7
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