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Projectile Motion The position of a projectile fired with an initial velocity feet per second and at an angle to the horizontal at the end of seconds is given by the parametric equations
See the illustration.
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a. Obtain the rectangular equation of the trajectory and identify the curve.
b. Show that the projectile hits the ground when .
c. How far has the projectile traveled (horizontally) when it strikes the ground? In other words, find the range .
d. Find the time when . Then find the horizontal distance and the vertical distance traveled by the projectile in this time. Then compute . This is the distance , the range, that the projectile travels up a plane inclined at to the horizontal . See the following illustration. (See also Problem 99 in Section 7.6.)
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To find:
a. The rectangular equation of the trajectory and identify the curve.
Answer to Problem 60AYU
a.
Explanation of Solution
Given:
The position of a projectile fired with an initial velocity feet per second and at an angle to the horizontal at the end of seconds is given by the parametric equations.
,
Formula used:
,
Calculation:
a. ,
is a quadratic function of and it represents the graph of a parabola where , and .
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To find:
b. To show that the projectile hits the ground when .
Answer to Problem 60AYU
b.
Explanation of Solution
Given:
The position of a projectile fired with an initial velocity feet per second and at an angle to the horizontal at the end of seconds is given by the parametric equations.
,
Formula used:
,
Calculation:
b.
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To find:
c. How far has the projectile traveled (horizontally) when it strikes the ground? In other words, find the range R.
Answer to Problem 60AYU
c.
Explanation of Solution
Given:
The position of a projectile fired with an initial velocity feet per second and at an angle to the horizontal at the end of seconds is given by the parametric equations.
,
Formula used:
,
Calculation:
c. To find R
We substitute in
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To find:
d. Find the time when . Then find the horizontal distance and the vertical distance traveled by the projectile in this time. Then compute . This is the distance R, the range, that the projectile travels up a plane inclined at to the horizontal .
Answer to Problem 60AYU
d. as
Explanation of Solution
Given:
The position of a projectile fired with an initial velocity feet per second and at an angle to the horizontal at the end of seconds is given by the parametric equations.
,
Formula used:
,
Calculation:
d.
;
At
hence,
as
Chapter 10 Solutions
Precalculus Enhanced with Graphing Utilities
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