At any instant the velocity of a particle of mass 500g is `(2t hati+ 3t^(2) hat j) ms^(-1)`.If the force acting on the particle at `t = 1s` is `(hat i+xhatj) N`. Then the value of x will be :

Options

A. `3`
B. `2`
C. `6`
D. `4`

Answer

To solve the problem, we need to find the value of \( x \) in the force vector \( \mathbf{F} = \hat{i} + x \hat{j} \) N, given the velocity of the particle and its mass.

### Step-by-Step Solution:

1. **Identify the given parameters:**
- Mass of the particle, \( m = 500 \, \text{g} = 0.5 \, \text{kg} \)
- Velocity of the particle, \( \mathbf{v}(t) = (2t) \hat{i} + (3t^2) \hat{j} \, \text{m/s} \)

2. **Calculate the acceleration:**
- Acceleration \( \mathbf{a} \) is the derivative of velocity \( \mathbf{v} \) with respect to time \( t \):
\[ \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d}{dt}[(2t) \hat{i} + (3t^2) \hat{j}] \]
- Differentiating each component:
\[ \mathbf{a} = (2) \hat{i} + (6t) \hat{j} \, \text{m/s}^2 \]

3. **Evaluate the acceleration at \( t = 1 \, \text{s} \):**
- Substitute \( t = 1 \):
\[ \mathbf{a}(1) = 2 \hat{i} + 6(1) \hat{j} = 2 \hat{i} + 6 \hat{j} \, \text{m/s}^2 \]

4. **Calculate the force acting on the particle:**
- Using Newton's second law, \( \mathbf{F} = m \mathbf{a} \):
\[ \mathbf{F} = 0.5 \, \text{kg} \cdot (2 \hat{i} + 6 \hat{j}) = (0.5 \cdot 2) \hat{i} + (0.5 \cdot 6) \hat{j} = 1 \hat{i} + 3 \hat{j} \, \text{N} \]

5. **Set the force equal to the given force vector:**
- We are given \( \mathbf{F} = \hat{i} + x \hat{j} \):
\[ 1 \hat{i} + 3 \hat{j} = \hat{i} + x \hat{j} \]

6. **Compare the coefficients:**
- From the \( \hat{j} \) component, we have:
\[ 3 = x \]

7. **Conclusion:**
- The value of \( x \) is \( 3 \).

### Final Answer:
The value of \( x \) is \( 3 \).