A particle has initial velocity `(2 vec(i)+3 vec(j))` and acceleration `(0.3 vec (i)+0.2 vec (j))`. The magnitude of velocity after 10 seconds will be

Options

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

Explanation

To find the magnitude of the velocity of a particle after a given time, we can use the equations of motion. Here's a step-by-step solution to the problem:

### Step 1: Identify the initial conditions
- **Initial velocity (U)**: \( \vec{U} = 2 \hat{i} + 3 \hat{j} \)
- **Acceleration (a)**: \( \vec{a} = 0.3 \hat{i} + 0.2 \hat{j} \)
- **Time (t)**: \( t = 10 \, \text{s} \)

### Step 2: Use the equation of motion
The equation of motion that relates initial velocity, acceleration, and final velocity is:
\[ \vec{V} = \vec{U} + \vec{a} t \]
Substituting the known values:
\[ \vec{V} = (2 \hat{i} + 3 \hat{j}) + (0.3 \hat{i} + 0.2 \hat{j}) \cdot 10 \]

### Step 3: Calculate the acceleration contribution
Calculate \( \vec{a} \cdot t \):
\[ \vec{a} \cdot t = (0.3 \hat{i} + 0.2 \hat{j}) \cdot 10 = (3 \hat{i} + 2 \hat{j}) \]

### Step 4: Substitute back into the equation for velocity
Now substitute this back into the equation for \( \vec{V} \):
\[ \vec{V} = (2 \hat{i} + 3 \hat{j}) + (3 \hat{i} + 2 \hat{j}) = (2 + 3) \hat{i} + (3 + 2) \hat{j} = 5 \hat{i} + 5 \hat{j} \]

### Step 5: Find the magnitude of the velocity
The magnitude of the velocity vector \( \vec{V} = 5 \hat{i} + 5 \hat{j} \) is given by:
\[ |\vec{V}| = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} \]

### Step 6: Simplify the magnitude
We can simplify \( \sqrt{50} \):
\[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \]

### Final Answer
Thus, the magnitude of the velocity after 10 seconds is:
\[ 5\sqrt{2} \, \text{units} \]