A particle has initial velocity `(2hati+3hatj)` and acceleration `(0.3hati+0.2hatj)`. The magnitude of velocity after 10 second will be

Options

A. `9 units`
B. `9sqrt(2) units`
C. `5sqrt(2) units`
D. `5 units`

Explanation

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

### Step 1: Identify the initial velocity and acceleration.
The initial velocity \( \mathbf{u} \) is given as:
\[ \mathbf{u} = 2\hat{i} + 3\hat{j} \]
The acceleration \( \mathbf{a} \) is given as:
\[ \mathbf{a} = 0.3\hat{i} + 0.2\hat{j} \]

### Step 2: Use the formula for final velocity.
The final velocity \( \mathbf{v} \) after time \( t \) can be calculated using the formula:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]
where \( t = 10 \) seconds.

### Step 3: Calculate the change in velocity due to acceleration.
First, calculate \( \mathbf{a} \cdot t \):
\[ \mathbf{a} \cdot t = (0.3\hat{i} + 0.2\hat{j}) \cdot 10 = (0.3 \cdot 10)\hat{i} + (0.2 \cdot 10)\hat{j} = 3\hat{i} + 2\hat{j} \]

### Step 4: Add the initial velocity and the change in velocity.
Now, substitute back into the equation for \( \mathbf{v} \):
\[ \mathbf{v} = (2\hat{i} + 3\hat{j}) + (3\hat{i} + 2\hat{j}) = (2 + 3)\hat{i} + (3 + 2)\hat{j} = 5\hat{i} + 5\hat{j} \]

### Step 5: Calculate the magnitude of the final velocity.
The magnitude \( |\mathbf{v}| \) of the velocity vector \( \mathbf{v} = 5\hat{i} + 5\hat{j} \) is given by:
\[ |\mathbf{v}| = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]

### Final Answer:
The magnitude of the velocity after 10 seconds is:
\[ 5\sqrt{2} \text{ meters per second} \]

---