To determine the acceleration of the couch, we need to apply the concepts of forces and Newton’s Second Law of Motion.
1. Identify the forces acting on the couch:
- Dale is pushing to the left with a force of [tex]\(100 \, \text{N}\)[/tex].
- Alex is pushing to the right with a force of [tex]\(175 \, \text{N}\)[/tex].
2. Determine the net force:
- Since the forces are in opposite directions, we subtract Dale’s force from Alex’s force.
[tex]\[
\text{Net Force} = \text{Force_{Alex}} - \text{Force_{Dale}}
\][/tex]
[tex]\[
\text{Net Force} = 175 \, \text{N} - 100 \, \text{N} = 75 \, \text{N} \, \text{(to the right)}
\][/tex]
3. Calculate the acceleration:
- Newton’s Second Law states that [tex]\( F = m \cdot a \)[/tex], where [tex]\( F \)[/tex] is the force, [tex]\( m \)[/tex] is the mass, and [tex]\( a \)[/tex] is the acceleration.
- We need to solve for [tex]\( a \)[/tex], which gives us [tex]\( a = \frac{F}{m} \)[/tex].
[tex]\[
a = \frac{\text{Net Force}}{\text{Mass}}
\][/tex]
[tex]\[
a = \frac{75 \, \text{N}}{55 \, \text{kg}}
\][/tex]
[tex]\[
a \approx 1.36 \, \text{m/s}^2
\][/tex]
Thus, the direction of the acceleration is to the right, since the net force is to the right.
Given the choices:
A. [tex]\(5 \, \text{m/s}^2\)[/tex] left
B. [tex]\(1.4 \, \text{m/s}^2\)[/tex] right
C. [tex]\(1.4 \, \text{m/s}^2\)[/tex] left
D. [tex]\(5 \, \text{m/s}^2\)[/tex] right
The best answer is:
B. [tex]\(1.4 \, \text{m/s}^2\)[/tex] right
