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Projectile Motion with Wind and Air Resistance Section 10.4b Practice Problems A baseball is hit when it is 3 ft above the ground. It leaves the bat with initial speed of 152 ft/sec, making an angle of 20 with the horizontal. At the instant the ball is hit, an instantaneous gust of wind blows in the horizontal direction directly opposite the direction the ball is taking toward the outfield, adding a component of 8.8i (ft/sec) to the ball’s initial velocity. (a) Find the vector equation (position vector) and the parametric equations for the path of the baseball. v 0 v0 cos i v0 sin j 8.8i 152 cos 20 i 152sin 20 j 8.8i 152 cos 20 8.8 i 152sin 20 j Initial position: r0 0i 3j Practice Problems A baseball is hit when it is 3 ft above the ground. It leaves the bat with initial speed of 152 ft/sec, making an angle of 20 with the horizontal. At the instant the ball is hit, an instantaneous gust of wind blows in the horizontal direction directly opposite the direction the ball is taking toward the outfield, adding a component of 8.8i (ft/sec) to the ball’s initial velocity. (a) Find the vector equation (position vector) and the parametric equations for the path of the baseball. 1 2 r gt j v 0t r0 2 2 16t j 152 cos 20 8.8 ti 152sin 20 tj 3j 152 cos 20 8.8 ti 3 152sin 20 t 16t 2 j Practice Problems A baseball is hit when it is 3 ft above the ground. It leaves the bat with initial speed of 152 ft/sec, making an angle of 20 with the horizontal. At the instant the ball is hit, an instantaneous gust of wind blows in the horizontal direction directly opposite the direction the ball is taking toward the outfield, adding a component of 8.8i (ft/sec) to the ball’s initial velocity. (a) Find the vector equation (position vector) and the parametric equations for the path of the baseball. r 152 cos 20 8.8 ti 3 152sin 20 t 16t Corresponding parametric equations: x 152 cos 20 8.8 t y 3 152sin 20 t 16t 2 2 j Practice Problems (b) How high does the baseball go, and when does it reach maximum height? y 3 152sin 20 t 16t dy 152sin 20 32t 0 dt 2 152sin 20 t 32 1.625sec 152sin 20 152sin 20 ymax 3 152sin 20 16 32 32 The baseball reaches a maximum 45.229ft height of about 45.229 ft after approximately 1.625 sec. 2 Practice Problems (c) Assuming the ball is not caught, find its range and flight time. y 3 152sin 20 t 16t 0 t 3.306sec 2 x R 152 cos 20 8.8 3.306 443.102ft The baseball is in flight for approximately 3.306 sec, and travels a horizontal distance of about 443.102 ft during that time. Confirm these answers graphically!!! Exploration 1 Suppose the baseball from Example 4 is hit toward the rightcenterfield fence, 400 ft from the batter. 1. If the fence is 15 ft high, is the hit a home run (does it clear the fence in flight)? Time for the ball to reach the fence: 152 cos 20 8.8 t 400 t 2.984sec Height of the ball at this time: y 3 152sin 20 2.984 16 2.984 15.647ft 2 The baseball clears the fence by less than a foot! Exploration 1 Suppose the baseball from Example 4 is hit toward the rightcenterfield fence, 400 ft from the batter. 2. Using graphical methods, find the range and the flight time if the angle at which the ball comes off the bat is: a 25, b 30, c 45 Graph in [0, 500] by [–20, 60] Angle (degrees) 25 30 45 Range (ft) 523.707 588.279 665.629 Flight time (sec) 4.061 4.789 6.745 Projectile Motion with Linear Drag The main force affecting the motion of a projectile, other than gravity, is air resistance. This slowing force is drag force and it acts in a direction opposite to the velocity of the projectile: Velocity Drag force Gravity Note: For projectiles moving through the air at relatively low speeds, the drag force is (very nearly) proportional to the speed (to the first power), and so is called linear. Projectile Motion with Linear Drag Equations for the motion of a projectile with linear drag force launched from the origin over a horizontal surface at t = 0: v0 kt Vector form: r 1 e cos i k g v0 kt kt 1 e sin 1 kt e j 2 k k v0 kt Parametric form: x 1 e cos k v0 g kt kt y 1 e sin 2 1 kt e k k where the drag coefficient k is a positive constant representing resistance due to air density. Exploration 2 Suppose the baseball from Example 4 is hit with linear drag to slow it down instead of wind. For the temperature and humidity on game day, the drag coefficient k is 0.05. 1. Find and graph the parametric equations for the path of the baseball. 152 0.05t x 1 e cos 20 0.05 152 0.05t y 3 1 e sin 20 0.05 32 0.05t 1 0.05t e 2 0.05 Graph in [0, 450] by [–20, 60] Exploration 2 Suppose the baseball from Example 4 is hit with linear drag to slow it down instead of wind. For the temperature and humidity on game day, the drag coefficient k is 0.05. 2. How high does the baseball go, and when does it reach this maximum height? Estimate answers graphically… The ball reaches a maximum height of about 43.07 ft when t is about 1.56 sec. 3. Assuming that the ball is not caught, find the range and flight time of the baseball. Estimate answers graphically… The range is about 425.47 ft and the flight time is about 3.23 sec.