A body with mass 5 kg is acted upon by a force `vec(F) = (- 3 hat (i) + 4 hat (j)) N`. If its initial velocity at t =0 is `vec(v) = 6 hat(i) - 12 hat (j) ms^(-1)`, the time at which it will just have a velocity along the y-axis is :

Options

A. never
B. 10 s
C. 2 s
D. 15 s

Answer

To solve the problem, we need to determine the time at which the velocity of the body will be entirely along the y-axis, meaning that the x-component of the velocity will be zero.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Mass of the body, \( m = 5 \, \text{kg} \)
- Force acting on the body, \( \vec{F} = -3 \hat{i} + 4 \hat{j} \, \text{N} \)
- Initial velocity, \( \vec{v_0} = 6 \hat{i} - 12 \hat{j} \, \text{ms}^{-1} \)

2. **Calculate the Acceleration:**
- The acceleration \( \vec{a} \) can be found using Newton's second law:
\[ \vec{a} = \frac{\vec{F}}{m} = \frac{-3 \hat{i} + 4 \hat{j}}{5} = -\frac{3}{5} \hat{i} + \frac{4}{5} \hat{j} \, \text{ms}^{-2} \]

3. **Determine the Components of Velocity:**
- The initial velocity components are:
- \( v_{0x} = 6 \, \text{ms}^{-1} \)
- \( v_{0y} = -12 \, \text{ms}^{-1} \)

4. **Use the Equation of Motion for the x-component:**
- The equation for the x-component of velocity is given by:
\[ v_x = v_{0x} + a_x t \]
- Since we want to find the time when \( v_x = 0 \):
\[ 0 = 6 - \frac{3}{5} t \]

5. **Solve for Time \( t \):**
- Rearranging the equation gives:
\[ \frac{3}{5} t = 6 \]
- Multiplying both sides by \( \frac{5}{3} \):
\[ t = 6 \times \frac{5}{3} = 10 \, \text{s} \]

6. **Conclusion:**
- The time at which the body will have a velocity entirely along the y-axis is \( t = 10 \, \text{s} \).

### Final Answer:
The time at which the body will just have a velocity along the y-axis is **10 seconds**.