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Notes 10.6 Application of Parametric Equations
Projectile motion, such as a ball being thrown into the air, can be modeled by parametric equations. If an object is thrown from a point [pic] feet above ground level with an initial speed
of [pic] ft/sec at an angle [pic] with the horizontal, the path of the object is given by the parametric
equations
| [pic] |
|[pic] = initial velocity or speed |
|[pic] = initial height |
The x-component represents the horizontal distance = (x-component of initial velocity)(time).
The y-component represents the vertical distance (height of the object) and comes from the projectile motion equation used in physics.
Ex. Kevin hits a baseball at 3 ft above the ground with an initial speed of 150 ft/sec at an angle
of 18° with the horizontal.
(a) Write the parametric equations that represent the motion of the baseball.
(b) Graph your equations from (a) on your graphing calculator, and sketch the graph below.
(c) What is the maximum height of the baseball? What is the time when it reaches its maximum
height? How far is this from Kevin in the horizontal direction?
(d) When will it hit the ground? How far is this from Kevin in the horizontal direction?
(e) Will it clear a 20-ft wall that is 400 ft away? Show the wall on your graph, and explain why
you answered “yes” or “no.”
WORKSHEET 10.6 Applications of Parametric Equations and More on Parametric Equations
Use the equations below to answer the following.
|[pic] |
|[pic] = initial velocity or speed, [pic] = initial height |
1. Jackie hits a baseball at 3.5 ft above the ground with an initial speed of 140 ft/sec at an angle
of 20° with the horizontal.
(a) Write the parametric equations that represent the motion of the baseball.
(b) Graph your equations from (a) on your graphing calculator, and sketch the graph below.
(c) What is the maximum height of the baseball? What is the time when it reaches its maximum
height? How far is this from Jackie in the horizontal direction?
(d) When will it hit the ground? How far is this from Jackie in the horizontal direction?
(e) Will it clear a 20-ft wall that is 350 ft away? Show the wall on your graph, and explain why you answered “yes” or “no.”
________________________________________________________________________________
2. Orlando hits a golf ball with an initial velocity of 180 ft/sec at and angle of 15° with the
horizontal.
(a) Write the parametric equations that represent the motion of the golf ball.
(b) Graph your equations from (a) on your graphing calculator, and sketch the graph below.
(c) What is the maximum height of the golf ball? What is the time when it reaches its maximum
height? How far is this from Jackie in the horizontal direction?
(d) When will it hit the ground? How far is this from Orlando in the horizontal direction?\
Use the equations below to answer the following.
|[pic] |
|[pic] = initial velocity or speed |
|[pic] = initial height |
3. Joey hits a baseball at 3.3 ft above the ground with an initial speed of 125 ft/sec at an angle
of 28° with the horizontal.
(a) Write the parametric equations that represent the motion of the baseball.
(b) What is the maximum height of the baseball? What is the time when it reaches its maximum
height? How far is this from Joey in the horizontal direction?
(c) When will it hit the ground? How far is this from Joey in the horizontal direction?
________________________________________________________________________________
Use the equations below to answer the following
|[pic] and [pic] |
|[pic]initial velocity or speed |
|[pic]initial height |
4. Robert hits a baseball at 3.2 ft above the ground with initial speed of 105 ft/sec at and angle of 23° with the horizontal.
a. Write the parametric equations that represent the motion of the baseball.
b. What is the maximum height of the baseball? What is the time when it reaches its maximum height? How far is this from Robert in the horizontal direction?
c. When will it hit the ground? How far is this from Robert in the horizontal direction?
(a) Make a table and graph, indicating the direction.
(b) Eliminate the parameter.
5. [pic]
6. [pic]
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y
x
y
x
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