12.3q5 ### Projectile Motion in Baler SystemsUnderstanding 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

Name: 12.3q5 ### Projectile Motion in Baler SystemsUnderstanding projectile
Uploaded: 2024-03-20T06:56:46.000Z
Channel: Bartleby
Description: 12.3q5 ### Projectile Motion in Baler SystemsUnderstanding 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 \