Parametric equations with projectiles The following equations are useful to model the x and y-coordinates of projectile motion launched at an angle θ (in degrees), initial velocity v 0 and acceleration due to gravity, g. The values (x 0,y 0 ) represent the starting location. x(t) = v 0 cos(θ)t + x 0 y(t) = v 0 sin(θ)t (1/2)gt 2 + y 0 g = 9.8m/s 2 g = 32ft/s 2 The value of g depends the units used in the problem statement! The following equations are useful in determining WINDOW values of the graphing calculator when modeling equations. height of projectile = v 0 2 sin 2 (θ)/2g flight time = 2v 0 sin(θ)/g projectile range = v 02 sin(2θ)/g Example 1 Several bottle rockets are fired with an initial velocity of 50 m/s from a starting location of (0, 0). Complete the table provided the various launch angles. Round your answers to the tenths. Initial Velocity Launch Angle Height (m) Range(m) Max flight time (sec) 50 25 50 30 50 45 50 50 50 60 Use trial and error to find BOTH launch angles that will have a maximum range of 100 meters. Use trial and error to find the launch angle that will have a maximum flight time of 10 seconds.
Example 2 - Several experimental bottle rockets are fired with various initial velocities from a starting location of (0, 0) and an initial angle of 45 0. Complete the table provided the various initial velocities. Round your answers to the tenths. Initial Velocity (m/sec) Launch Angle Height (m) Range (m) Max flight time (sec) 20 45 30 45 40 45 60 45 80 45 Use trial and error to the initial velocity that will have a maximum range of 4500 meters.
Example 3 - Biff the concert promoter used a T-shirt cannon to shoot some T-shirts into open windows of a college dorm. He is on a platform 60 feet from the dorm and 10 feet above the ground. The T-shirt cannon is fired at an angle of 32 0 and has an initial velocity of 90 ft/sec. 3a) What are the parametric equations for the situation? 3b) What are good values to use for your WINDOW parameters? 3c) Each floor of the dorm is 10ft tall. The bottom 3feet are made up of a wall while the top 7 feet are large windows. Draw a rough sketch of the problem situation showing at least 4 floors of the dorm. 3d) Determine the floor that will receive the T-shirt 3e) At what angle should he aim his T-shirt cannon to deliver T-shirts to the 5 th floor?
Example 4 Alex is 14 feet directly in front of a soccer goal. Alex is 5.5 feet tall with a maximum vertical jump of 3 feet. He jumps up 3 feet and tries to hit the ball with his head directly toward the net at a velocity of 30 ft/sec. The diameter of a soccer ball is approximately 9 inches. 4a) What are the parametric equations for the situation? 4b) What are good values to use for your WINDOW parameters? 4c) At what angle must Alex hit the ball in order for it to pass directly under the cross bar of the goal which is 8 feet above the ground? Jimmie is on the opposite team. He is directly in front of the goal and in line with Alex. Jimmie is only 5 feet tall and tries to block Alex's shot with his head by jumping up at a velocity of 2.5 ft/sec. 4d) Create a set of equations that model Jimmie's attempt to block the goal. 4e) Create a simple table of 4 rows that shows the location of the ball and Jimmie's head as the ball is near the goal. Use your table to justify if Alex scores a goal or not. T Alex X Alex Y Jimmie X Jimmie Y 0.47 0.48 049 0.50
Example 5 Maggie is playing miniature golf. She must hit the ball so that it rolls up a ramp that has an angle of inclination of 38 0. She wants the ball to travel through a 2 foot tall hole in a wall that is 15 feet away. The bottom of the hole is 5 feet from the ground. 5a) Draw a simple sketch of the situation with all described dimensions. 5b) What are good values to use for your WINDOW parameters? 5c) What are the parametric equations that show Maggie successfully shooting the ball through the hole in the wall? 5d) If the ramp is moved 30 feet from the wall, what are good values to use for your WINDOW parameters? 5e) What are the parametric equations that show Maggie successfully shooting the ball through the hole in the wall if the ramp was 30 feet from the wall?
Example 6 Freddy the frog wants to jump to another lily pad that is 6 feet away. Each lily pad is a foot diameter. He leaps at an angle of 35 0 toward the other lily pad. At the same time Freddy jumps, a largemouth bass located directly between the lily pads jumps up at a rate of 6 ft/sec. 6a) Draw a simple sketch of the situation with all described dimensions. 6b) What are good values to use for your WINDOW parameters? 6c) What are the parametric equations that model Freddy's attempted path and the path of the largemouth bass? 6d) Freddy makes it to the other lily pad! Use the TRACE function on your calculator to determine the time of intersection to the nearest hundredths.
Example 7 A projectile is launched at an initial velocity of 110 feet/sec from a platform 22 feet above the ground. Determine the launch angle needed to reach the indicated height at a horizontal distance of 75 feet from the launch point. x(t) = v 0 cos(θ)t + x 0 y(t) = v 0 sin(θ)t (1/2)gt 2 + y 0 g = 9.8m/s 2 g = 32ft/s 2 The value of g depends the units used in the problem statement! Angle (degrees) Height (feet) 35 45 60 75 90 Example 8 A projectile is launched from a platform that is 22 meters above the ground at an angle of 40 0. Determine the approximate launch velocity needed to reach the indicated height at a horizontal distance of 120 meters from the launch point. x(t) = v 0 cos(θ)t + x 0 y(t) = v 0 sin(θ)t (1/2)gt 2 + y 0 g = 9.8m/s 2 g = 32ft/s 2 The value of g depends the units used in the problem statement! Velocity (m/s) Height (meters) 20 50 60 80 100