9.8: Modeling Motion Using Parametric Equations

Objectives Model the motion of a projectile using parametric equations. Solve problems related to the motion of a projectile, its trajectory, and range.

Real World Application Suppose a professional football player kicks a football with an initial velocity of 29 yards per second at an angle of 68° to the horizontal. Suppose a kick returner catches the ball 5 seconds later. How far has the ball traveled horizontally and what is its vertical height at the time?

Real World Application

Vocabulary Projectile: object that is launched (football) Trajectory: path of the projectile Range: horizontal distance that a projectile travels

Trajectory Initial trajectory of a punted football θ

Components As the ball moves, gravity will act on it in the vertical direction. The horizontal component is not affected by gravity. If we ignore air resistance, the horizontal speed is constant. Range

Continued . . . . The vertical component of the velocity of the ball is large and positive at the beginning, and decreases to zero at the top.

Continued . . . . The vertical component then increases in the negative direction as it falls. When it reaches the ground, the vertical speed is the same as when it left the kicker’s foot, but in the opposite directions.

Parametric Equations Parametric equations can represent the position of the ball relative to the starting point in terms of the parameter time. θ

Example Find the initial horizontal velocity and vertical velocity of a stone kicked with an initial velocity of 18 ft/sec at an angle 37° with the ground.

Horizontal Distance Horizontal distance = horizontal velocity × time (Remember that the horizontal component is not affected by gravity.)

Vertical Displacement Displacement due to initial velocity Vertical displacement Displacement due to gravity = –

Example A punter kicks the ball with an initial velocity of 29 yd/sec at an angle 68° to the horizontal. The returner catches the ball 5 seconds later. How far has the ball traveled horizontally? What is its vertical height at that time? (Since gravity is ft/sec2: 29yd/sec×3ft/yd=87 ft/sec)

Example Continued Suppose the kick returner lets the ball hit the ground. What is the hang time (elapsed time between the moment the ball is kicked and the time it hits the ground? (Find t when y=0.) About 5 seconds

Initial Height The object may not be launched from ground level. A baseball may be hit at a height of 3 feet. In this case, you need to add the initial vertical height to y.

Example A softball pitcher throws the ball at an angle of 5.2° with the horizontal at a speed of 67 mph. The distance from the pitcher’s mound to home plate is 43 feet. If the ball is released 2.7 feet above ground, how far above ground is the ball when it crosses home plate?

Solution

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