Use the model for projectile motion, assuming there is no air resistance and g = 32 feet per second per second. A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector. (a) Find the minimum initial speed of the bale, and the corresponding angle, 8, at which it must be ejected from the baler. (Round your answers to two decimal places.) ft/sec (b) The ejector has a fixed angle of 45°. Find the initial speed, v, required. ft/sec
Use the model for projectile motion, assuming there is no air resistance and g = 32 feet per second per second. A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector. (a) Find the minimum initial speed of the bale, and the corresponding angle, 8, at which it must be ejected from the baler. (Round your answers to two decimal places.) ft/sec (b) The ejector has a fixed angle of 45°. Find the initial speed, v, required. ft/sec
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![### Projectile Motion in Baler Systems
Understanding projectile motion is essential in optimizing the functionality of agricultural machinery like balers. Assuming no air resistance and using the standard acceleration due to gravity \( g = 32 \) feet per second squared (\( \text{ft/s}^2 \)), the following example illustrates how to apply this model.
A **bale ejector** consists of two variable-speed belts at the end of a baler, designed to toss bales into a trailing wagon. To load the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector.
#### Problem Statement:
(a) **Find the minimum initial speed ( \( v \) ) and corresponding angle ( \( \theta \) )**
at which the bale must be ejected to reach the specified position (8 feet above, 16 feet behind).
\[
v = \boxed{\text{ ft/sec}}
\]
\[
\theta = \boxed{\text{ °}}
\]
(b) **Given a fixed angle of \( 45^\circ \)**, find the required initial speed ( \( v \) ).
\[
v = \boxed{\text{ ft/sec}}
\]
### Explanation:
1. **Setting Up the Equations:**
The equations of projectile motion describe the horizontal (\( x \)) and vertical (\( y \)) displacement of the bale. Let:
\[
x = v \cos(\theta) \cdot t
\]
\[
y = v \sin(\theta) \cdot t - \frac{1}{2}gt^2
\]
2. **Given Values:**
\[
x = 16 \text{ feet}, \quad y = 8 \text{ feet}, \quad g = 32 \text{ ft/s}^2
\]
3. **Solving for Minimum Initial Speed and Angle:**
Substitute \( x \) and \( y \) into their respective equations and solve for \( v \) and \( \theta \).
4. **Fixed Angle Scenario:**
For \( \theta = 45^\circ \):
\[
x = v \cos(45^\circ) \cdot t
\]
\[
y = v \sin(45^\circ) \cd](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafc8379c-b6a2-4db8-9fce-fab4471db5da%2F111ed0e1-41f6-4750-8a30-9e0fd41237dd%2F7kl3z9l_processed.png&w=3840&q=75)
Transcribed Image Text:### Projectile Motion in Baler Systems
Understanding projectile motion is essential in optimizing the functionality of agricultural machinery like balers. Assuming no air resistance and using the standard acceleration due to gravity \( g = 32 \) feet per second squared (\( \text{ft/s}^2 \)), the following example illustrates how to apply this model.
A **bale ejector** consists of two variable-speed belts at the end of a baler, designed to toss bales into a trailing wagon. To load the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector.
#### Problem Statement:
(a) **Find the minimum initial speed ( \( v \) ) and corresponding angle ( \( \theta \) )**
at which the bale must be ejected to reach the specified position (8 feet above, 16 feet behind).
\[
v = \boxed{\text{ ft/sec}}
\]
\[
\theta = \boxed{\text{ °}}
\]
(b) **Given a fixed angle of \( 45^\circ \)**, find the required initial speed ( \( v \) ).
\[
v = \boxed{\text{ ft/sec}}
\]
### Explanation:
1. **Setting Up the Equations:**
The equations of projectile motion describe the horizontal (\( x \)) and vertical (\( y \)) displacement of the bale. Let:
\[
x = v \cos(\theta) \cdot t
\]
\[
y = v \sin(\theta) \cdot t - \frac{1}{2}gt^2
\]
2. **Given Values:**
\[
x = 16 \text{ feet}, \quad y = 8 \text{ feet}, \quad g = 32 \text{ ft/s}^2
\]
3. **Solving for Minimum Initial Speed and Angle:**
Substitute \( x \) and \( y \) into their respective equations and solve for \( v \) and \( \theta \).
4. **Fixed Angle Scenario:**
For \( \theta = 45^\circ \):
\[
x = v \cos(45^\circ) \cdot t
\]
\[
y = v \sin(45^\circ) \cd
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