An object is acted upon by the forces `vec(F)_(1)=4hat(i)N` and `vec(F)_(2)=(hat(i)-hat(j))N`. If the displacement of the object is `D=(hat(i)+6hat(j)-hat(k))`m, the kinetic energy of the object

Options

A. remains constant
B. increases by 1 J
C. decreases by 1 J
D. decreases by 2 J

Answer

To solve the problem, we need to determine the change in kinetic energy of the object acted upon by the given forces and displacement. We will use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.

### Step-by-Step Solution:

1. **Identify the Forces**:
- The first force is given as \( \vec{F}_1 = 4 \hat{i} \, \text{N} \).
- The second force is given as \( \vec{F}_2 = \hat{i} - \hat{j} \, \text{N} \).

2. **Calculate the Net Force**:
- The net force \( \vec{F}_{\text{net}} \) acting on the object is the vector sum of \( \vec{F}_1 \) and \( \vec{F}_2 \):
\[ \vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 = (4 \hat{i}) + (\hat{i} - \hat{j}) = (4 + 1) \hat{i} - \hat{j} = 5 \hat{i} - \hat{j} \, \text{N} \]

3. **Identify the Displacement**:
- The displacement \( \vec{D} \) is given as \( \vec{D} = \hat{i} + 6 \hat{j} - \hat{k} \, \text{m} \).

4. **Calculate the Work Done**:
- The work done \( W \) by the net force on the object is given by the dot product of the net force and the displacement:
\[ W = \vec{F}_{\text{net}} \cdot \vec{D} = (5 \hat{i} - \hat{j}) \cdot (\hat{i} + 6 \hat{j} - \hat{k}) \]
- Calculate the dot product:
\[ W = 5 \cdot 1 + (-1) \cdot 6 + 0 = 5 - 6 + 0 = -1 \, \text{J} \]

5. **Apply the Work-Energy Theorem**:
- According to the work-energy theorem, the work done is equal to the change in kinetic energy:
\[ W = \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \]
- From our calculation, we have:
\[ -1 \, \text{J} = KE_{\text{final}} - KE_{\text{initial}} \]

6. **Determine the Final Kinetic Energy**:
- Rearranging the equation gives:
\[ KE_{\text{final}} = KE_{\text{initial}} - 1 \, \text{J} \]
- This shows that the final kinetic energy is 1 Joule less than the initial kinetic energy.

### Final Answer:
The kinetic energy of the object decreases by 1 Joule.