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

Explanation

To solve the problem step by step, we need to analyze the motion of the body under the influence of the given force.

### Step 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 at \( t = 0 \), \( \vec{v}_0 = (6 \hat{i} - 12 \hat{j}) \, \text{m/s} \)

### Step 2: Calculate the Acceleration
Using Newton's second law, \( \vec{F} = m \vec{a} \), we can find the acceleration \( \vec{a} \):
\[ \vec{a} = \frac{\vec{F}}{m} = \frac{-3 \hat{i} + 4 \hat{j}}{5} = \left(-\frac{3}{5} \hat{i} + \frac{4}{5} \hat{j}\right) \, \text{m/s}^2 \]

### Step 3: Determine the Velocity Components
The velocity of the body at any time \( t \) can be expressed using the equation:
\[ \vec{v} = \vec{v}_0 + \vec{a} t \]
Breaking this down into components:
- Velocity in the x-direction:
\[ v_x = v_{0x} + a_x t = 6 - \frac{3}{5} t \]
- Velocity in the y-direction:
\[ v_y = v_{0y} + a_y t = -12 + \frac{4}{5} t \]

### Step 4: Find the Time When Velocity Along the x-axis is Zero
We want to find the time \( t \) when the velocity along the x-axis becomes zero:
\[ 0 = 6 - \frac{3}{5} t \]
Rearranging gives:
\[ \frac{3}{5} t = 6 \]
\[ t = 6 \cdot \frac{5}{3} = 10 \, \text{s} \]

### Step 5: Conclusion
At \( t = 10 \, \text{s} \), the velocity along the x-axis becomes zero. Thus, the body will have a velocity along the y-axis only at this time.

### Final Answer
The time at which the body will just have a velocity along the y-axis is \( t = 10 \, \text{s} \).

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