A particle has an initial velocity `3hati+4hatj` and an accleration of `0.4hati+0.3hatj`. Its speed after 10s is

Options

A. 10 unit
B. 7 units
C. `7sqrt(2)` units
D. `8.5` units

Answer

To find the speed of a particle after 10 seconds given its initial velocity and acceleration, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial velocity \( \mathbf{u} = 3\hat{i} + 4\hat{j} \)
- Acceleration \( \mathbf{a} = 0.4\hat{i} + 0.3\hat{j} \)
- Time \( t = 10 \, \text{s} \)

2. **Use the Equation of Motion:**
We will use the equation of motion for velocity:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} t \]

3. **Substitute the Values:**
Substitute the values of \( \mathbf{u} \), \( \mathbf{a} \), and \( t \):
\[ \mathbf{v} = (3\hat{i} + 4\hat{j}) + (0.4\hat{i} + 0.3\hat{j}) \times 10 \]

4. **Calculate the Acceleration Contribution:**
Calculate \( \mathbf{a} t \):
\[ \mathbf{a} t = (0.4\hat{i} + 0.3\hat{j}) \times 10 = 4\hat{i} + 3\hat{j} \]

5. **Add the Initial Velocity and the Acceleration Contribution:**
Now, add \( \mathbf{u} \) and \( \mathbf{a} t \):
\[ \mathbf{v} = (3\hat{i} + 4\hat{j}) + (4\hat{i} + 3\hat{j}) = (3 + 4)\hat{i} + (4 + 3)\hat{j} = 7\hat{i} + 7\hat{j} \]

6. **Find the Magnitude of the Final Velocity:**
The speed is the magnitude of the velocity vector \( \mathbf{v} \):
\[ |\mathbf{v}| = \sqrt{(7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]

### Final Answer:
The speed of the particle after 10 seconds is \( 7\sqrt{2} \, \text{m/s} \).