A particle has initial velocity `(3 hat i+ 4 hat j)` and has acceleration `(0.1 hat i+ 0.3 hat j)`. Its speed after `10 s` is.

Options

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

Explanation

To find the speed of the particle after 10 seconds, we can follow these steps:

### Step 1: Identify the initial velocity and acceleration
The initial velocity \( \mathbf{u} \) is given as:
\[ \mathbf{u} = 3 \hat{i} + 4 \hat{j} \]
The acceleration \( \mathbf{a} \) is given as:
\[ \mathbf{a} = 0.1 \hat{i} + 0.3 \hat{j} \]

### Step 2: Use the formula for final velocity
The final velocity \( \mathbf{v} \) after time \( t \) can be calculated using the equation:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]
Substituting the values we have:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (0.1 \hat{i} + 0.3 \hat{j}) \cdot 10 \]

### Step 3: Calculate the acceleration term
Calculating the acceleration term:
\[ \mathbf{a} \cdot t = (0.1 \hat{i} + 0.3 \hat{j}) \cdot 10 = (0.1 \cdot 10) \hat{i} + (0.3 \cdot 10) \hat{j} = 1 \hat{i} + 3 \hat{j} \]

### Step 4: Substitute back to find final velocity
Now substituting back into the equation for final velocity:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (1 \hat{i} + 3 \hat{j}) = (3 + 1) \hat{i} + (4 + 3) \hat{j} = 4 \hat{i} + 7 \hat{j} \]

### Step 5: Calculate the speed
The speed is the magnitude of the velocity vector \( \mathbf{v} \):
\[ |\mathbf{v}| = \sqrt{(4)^2 + (7)^2} = \sqrt{16 + 49} = \sqrt{65} \]

### Step 6: Final answer
Thus, the speed of the particle after 10 seconds is:
\[ \sqrt{65} \text{ m/s} \]