A particle has an initial velocity `3hati+3hatj` and acceleration of `0.41hati+0.3hatj`. Its speed after 10s is

Options

A. 10 units
B. `7sqrt2` units
C. 7 units
D. 8.5 units

Answer

To solve the problem, we need to find the speed of a particle after 10 seconds given its initial velocity and acceleration.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial velocity \( \mathbf{u} = 3\hat{i} + 3\hat{j} \) m/s
- Acceleration \( \mathbf{a} = 0.41\hat{i} + 0.3\hat{j} \) m/s²
- Time \( t = 10 \) s

2. **Use the Equation of Motion:**
The equation to find the final velocity \( \mathbf{v} \) is given by:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]

3. **Substitute the Values:**
Substitute the values of \( \mathbf{u} \), \( \mathbf{a} \), and \( t \) into the equation:
\[ \mathbf{v} = (3\hat{i} + 3\hat{j}) + (0.41\hat{i} + 0.3\hat{j}) \cdot 10 \]

4. **Calculate the Acceleration Contribution:**
Calculate \( \mathbf{a} \cdot t \):
\[ \mathbf{a} \cdot t = (0.41\hat{i} + 0.3\hat{j}) \cdot 10 = 4.1\hat{i} + 3.0\hat{j} \]

5. **Combine the Vectors:**
Now add \( \mathbf{u} \) and \( \mathbf{a} \cdot t \):
\[ \mathbf{v} = (3\hat{i} + 3\hat{j}) + (4.1\hat{i} + 3.0\hat{j}) = (3 + 4.1)\hat{i} + (3 + 3.0)\hat{j} = 7.1\hat{i} + 6.0\hat{j} \]

6. **Calculate the Magnitude of the Velocity:**
The speed (magnitude of velocity) is given by:
\[ |\mathbf{v}| = \sqrt{(7.1)^2 + (6.0)^2} \]
\[ |\mathbf{v}| = \sqrt{50.41 + 36} = \sqrt{86.41} \approx 9.3 \text{ m/s} \]

### Final Answer:
The speed of the particle after 10 seconds is approximately \( 9.3 \) m/s.