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. 10s
C. 2s
D. 15s

Explanation

To solve the problem step by step, we will analyze the motion of the body under the influence of the given force and find the time when the velocity is directed along the y-axis.

### 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, \( \vec{v_0} = 6 \hat{i} - 12 \hat{j} \, \text{m/s} \)

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

### Step 3: Write the Velocity Equation
The velocity \( \vec{v} \) at time \( t \) can be expressed as:
\[ \vec{v} = \vec{v_0} + \vec{a} t \]
Substituting the initial velocity and acceleration:
\[ \vec{v} = (6 \hat{i} - 12 \hat{j}) + \left(-\frac{3}{5} \hat{i} + \frac{4}{5} \hat{j}\right) t \]
This simplifies to:
\[ \vec{v} = \left(6 - \frac{3}{5} t\right) \hat{i} + \left(-12 + \frac{4}{5} t\right) \hat{j} \]

### Step 4: Set the x-component of Velocity to Zero
For the velocity to be directed along the y-axis, the x-component of the velocity must be zero:
\[ 6 - \frac{3}{5} t = 0 \]

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

### Final Answer
The time at which the body will just have a velocity along the y-axis is:
\[ \boxed{10 \, \text{s}} \]