The initial velocity of a 4.0-kg box is 11 m/s , due west. After the box slides 4.0 m horizontally, its speed is 1.5 m/s. Determine the magnitude and the direction of the non-conservative force acting on the box as it slides .

Options

A. 42N,duewest
B. 31N,dueeast
C. 83N,duewest
D. 59N,dueeast

Answer

To solve the problem, we need to determine the magnitude and direction of the non-conservative force acting on the box as it slides. We can do this using the work-energy principle, which states that the work done by non-conservative forces is equal to the change in kinetic energy of the system.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of the box, \( m = 4.0 \, \text{kg} \)
- Initial velocity, \( u = 11 \, \text{m/s} \) (due west)
- Final velocity, \( v = 1.5 \, \text{m/s} \)
- Distance slid, \( s = 4.0 \, \text{m} \)

2. **Calculate Initial and Final Kinetic Energy:**
- Initial kinetic energy, \( KE_i = \frac{1}{2} m u^2 = \frac{1}{2} \times 4.0 \, \text{kg} \times (11 \, \text{m/s})^2 \)
- \( KE_i = 2.0 \times 121 = 242 \, \text{J} \)

- Final kinetic energy, \( KE_f = \frac{1}{2} m v^2 = \frac{1}{2} \times 4.0 \, \text{kg} \times (1.5 \, \text{m/s})^2 \)
- \( KE_f = 2.0 \times 2.25 = 4.5 \, \text{J} \)

3. **Calculate Change in Kinetic Energy:**
- Change in kinetic energy, \( \Delta KE = KE_f - KE_i = 4.5 \, \text{J} - 242 \, \text{J} = -237.5 \, \text{J} \)

4. **Relate Work Done to Change in Kinetic Energy:**
- According to the work-energy principle, the work done by non-conservative forces is equal to the change in kinetic energy:
\[ W = \Delta KE \]
- Therefore, \( W = -237.5 \, \text{J} \)

5. **Calculate Work Done by Non-Conservative Forces:**
- The work done can also be expressed as:
\[ W = F \cdot s \]
- Here, \( F \) is the magnitude of the non-conservative force and \( s = 4.0 \, \text{m} \).
- Thus, we have:
\[ F \cdot 4.0 \, \text{m} = -237.5 \, \text{J} \]
- Rearranging gives:
\[ F = \frac{-237.5 \, \text{J}}{4.0 \, \text{m}} = -59.375 \, \text{N} \]

6. **Determine the Direction of the Force:**
- The negative sign indicates that the direction of the force is opposite to the direction of motion of the box (which is due west).
- Therefore, the non-conservative force is acting towards the east.

### Final Answer:
- **Magnitude of the Non-Conservative Force:** \( 59.375 \, \text{N} \)
- **Direction of the Non-Conservative Force:** Due east