A particle moves in plane with acceleration `veca = 2hati+2hatj`. If `vecv=3hati+6hatj`, then final velocity of particle at time `t=2s` is

Options

A. `5hati+8hatj`
B. `7hati+10hatj`
C. `2hati+6hatj`
D. `3hati+2hatj`

Answer

To find the final velocity of the particle at time \( t = 2 \, \text{s} \), we will use the kinematic equation:

\[ \vec{v} = \vec{u} + \vec{a} t \]

Where:
- \( \vec{v} \) is the final velocity,
- \( \vec{u} \) is the initial velocity,
- \( \vec{a} \) is the acceleration, and
- \( t \) is the time.

### Step 1: Identify the given values
From the problem, we have:
- Initial velocity \( \vec{u} = 3 \hat{i} + 6 \hat{j} \)
- Acceleration \( \vec{a} = 2 \hat{i} + 2 \hat{j} \)
- Time \( t = 2 \, \text{s} \)

### Step 2: Substitute the values into the equation
Now we will substitute the values into the equation:

\[ \vec{v} = (3 \hat{i} + 6 \hat{j}) + (2 \hat{i} + 2 \hat{j}) \cdot 2 \]

### Step 3: Calculate the acceleration term
First, we need to calculate \( \vec{a} \cdot t \):

\[ \vec{a} \cdot t = (2 \hat{i} + 2 \hat{j}) \cdot 2 = (2 \cdot 2) \hat{i} + (2 \cdot 2) \hat{j} = 4 \hat{i} + 4 \hat{j} \]

### Step 4: Add the initial velocity and the acceleration term
Now, we add the initial velocity \( \vec{u} \) and the result from the previous step:

\[ \vec{v} = (3 \hat{i} + 6 \hat{j}) + (4 \hat{i} + 4 \hat{j}) \]

### Step 5: Combine like terms
Combining the \( \hat{i} \) and \( \hat{j} \) components:

\[ \vec{v} = (3 + 4) \hat{i} + (6 + 4) \hat{j} = 7 \hat{i} + 10 \hat{j} \]

### Final Answer
Thus, the final velocity of the particle at \( t = 2 \, \text{s} \) is:

\[ \vec{v} = 7 \hat{i} + 10 \hat{j} \]

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