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 the particle after 10 seconds, we can follow these steps:

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

### Step 2: Use the equation of motion to find the final velocity
We can use the first equation of motion:
\[ \vec{v} = \vec{u} + \vec{a} \cdot t \]
where \( t = 10 \) seconds.

Substituting the values:
\[ \vec{v} = (2 \hat{i} + 3 \hat{j}) + (0.3 \hat{i} + 0.2 \hat{j}) \cdot 10 \]

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

### Step 4: Add the initial velocity and the acceleration component
Now, substituting back into the equation for velocity:
\[ \vec{v} = (2 \hat{i} + 3 \hat{j}) + (3 \hat{i} + 2 \hat{j}) = (2 + 3) \hat{i} + (3 + 2) \hat{j} = 5 \hat{i} + 5 \hat{j} \]

### Step 5: Calculate the magnitude of the velocity
The magnitude of the velocity \( |\vec{v}| \) is given by:
\[ |\vec{v}| = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} \]

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

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

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