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. 0
B. 10s
C. 2s
D. 15s

Explanation

To solve the problem, we need to determine the time at which the velocity of the body will be directed only along the y-axis. This means that the x-component of the velocity must be zero.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- 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, \( \vec{v}_0 = 6 \hat{i} - 12 \hat{j} \, \text{m/s} \)

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

3. **Determine the Velocity Components:**
- The initial velocity in the x-direction, \( v_{0x} = 6 \, \text{m/s} \)
- The acceleration in the x-direction, \( a_x = -\frac{3}{5} \, \text{m/s}^2 \)
- The velocity in the x-direction at time \( t \) is given by:
\[ v_x(t) = v_{0x} + a_x t = 6 - \frac{3}{5} t \]

4. **Set the x-velocity to Zero:**
- To find the time when the velocity is only along the y-axis, set \( v_x(t) = 0 \):
\[ 0 = 6 - \frac{3}{5} t \]

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

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

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