Projectile Motion An object is propelled upward at an angle θ , 45 ∘ < θ < 90 ∘ , to the horizontal with an initial velocity v 0 feet per second from the base of a plane that makes an angle of 45 ∘ with the horizontal. See the illustration. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the function R ( θ ) = v 0 2 2 16 cos θ ( sin θ − cos θ ) Show that R ( θ ) = v 0 2 2 32 [ s i n ( 2 θ ) − c o s ( 2 θ ) − 1 ] In calculus, you will be asked to find the angle θ that maximizes R by solving the equation sin ( 2 θ ) + cos ( 2 θ ) = 0 solve the equation for θ . What is the maximum distance R if v 0 =32 feet per second? Graph R = R ( θ ) , 45 ∘ ≤ θ ≤ 90 ∘ , and find the angle θ that maximizes the distance R . Also find the maximum distance. Use v 0 = 32 feet per second. Compare the results with the answers found in parts (b) and (c).
Projectile Motion An object is propelled upward at an angle θ , 45 ∘ < θ < 90 ∘ , to the horizontal with an initial velocity v 0 feet per second from the base of a plane that makes an angle of 45 ∘ with the horizontal. See the illustration. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the function R ( θ ) = v 0 2 2 16 cos θ ( sin θ − cos θ ) Show that R ( θ ) = v 0 2 2 32 [ s i n ( 2 θ ) − c o s ( 2 θ ) − 1 ] In calculus, you will be asked to find the angle θ that maximizes R by solving the equation sin ( 2 θ ) + cos ( 2 θ ) = 0 solve the equation for θ . What is the maximum distance R if v 0 =32 feet per second? Graph R = R ( θ ) , 45 ∘ ≤ θ ≤ 90 ∘ , and find the angle θ that maximizes the distance R . Also find the maximum distance. Use v 0 = 32 feet per second. Compare the results with the answers found in parts (b) and (c).
Solution Summary: The author illustrates how an object is propelled upward at an angle of 45, to the horizontal with an initial velocity of v 0 feet per second.
Projectile Motion An object is propelled upward at an angle
, to the horizontal with an initial velocity
feet per second from the base of a plane that makes an angle of
with the horizontal. See the illustration. If air resistance is ignored, the distance
that it travels up the inclined plane is given by the function
Show that
In calculus, you will be asked to find the angle
that maximizes
by solving the equation
solve the equation for
.
What is the maximum distance
if
feet per second?
Graph
, and find the angle
that maximizes the distance
. Also find the maximum distance. Use
feet per second. Compare the results with the answers found in parts (b) and (c).
Expert Solution
To determine
To show:
a.
Answer to Problem 97AYU
a.
Explanation of Solution
Given:
An object is propelled upward at an angle , , to the horizontal with an initial velocity of feet per second from the base of a plane that makes an angle of with the horizontal. See the illustration. If air resistance is ignored, the distance that it travels up the inclined plane is given by the function .
Calculation:
a.
Expert Solution
To determine
To show:
b. In calculus, you will be asked to find the angle that maximizes by solving the equation . Solve this equation for .
Answer to Problem 97AYU
b.
Explanation of Solution
Given:
An object is propelled upward at an angle , , to the horizontal with an initial velocity of feet per second from the base of a plane that makes an angle of with the horizontal. See the illustration. If air resistance is ignored, the distance that it travels up the inclined plane is given by the function .
Calculation:
b.
Expert Solution
To determine
To show:
c. What is the maximum distance if feet per second?
Answer to Problem 97AYU
c.
Explanation of Solution
Given:
An object is propelled upward at an angle , , to the horizontal with an initial velocity of feet per second from the base of a plane that makes an angle of with the horizontal. See the illustration. If air resistance is ignored, the distance that it travels up the inclined plane is given by the function .
Calculation:
c.
Expert Solution
To determine
To show:
d. Graph , and find the angle that maximizes the distance . Also find the maximum distance. Use feet per second. Compare the results with the answers found in parts (b) and (c).
Answer to Problem 97AYU
d.
Explanation of Solution
Given:
An object is propelled upward at an angle , , to the horizontal with an initial velocity of feet per second from the base of a plane that makes an angle of with the horizontal. See the illustration. If air resistance is ignored, the distance that it travels up the inclined plane is given by the function .
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