A particle velocity changes from `(2hat(i)+3hat(j))ms^(-1)` to `(2hat(i)-3hat(j))ms^(-1)` in 2 s. The acceleration in `ms^(-2)` is

Options

A. `-(hat(i)+5hat(j))`
B. `(hat(i)+5hat(j))//2`
C. zero
D. `(-3hat(j))`

Explanation

To find the acceleration of the particle, we can follow these steps:

### Step 1: Identify the initial and final velocities
The initial velocity \( \mathbf{u} \) is given as:
\[ \mathbf{u} = 2\hat{i} + 3\hat{j} \, \text{m/s} \]
The final velocity \( \mathbf{v} \) is given as:
\[ \mathbf{v} = 2\hat{i} - 3\hat{j} \, \text{m/s} \]

### Step 2: Calculate the change in velocity
The change in velocity \( \Delta \mathbf{v} \) can be calculated using the formula:
\[ \Delta \mathbf{v} = \mathbf{v} - \mathbf{u} \]
Substituting the values:
\[ \Delta \mathbf{v} = (2\hat{i} - 3\hat{j}) - (2\hat{i} + 3\hat{j}) \]

### Step 3: Simplify the change in velocity
Now, we simplify the expression:
\[ \Delta \mathbf{v} = 2\hat{i} - 3\hat{j} - 2\hat{i} - 3\hat{j} = 0\hat{i} - 6\hat{j} = -6\hat{j} \, \text{m/s} \]

### Step 4: Calculate the acceleration
Acceleration \( \mathbf{a} \) is defined as the change in velocity divided by the time taken. The time \( t \) is given as 2 seconds. Thus, we can use the formula:
\[ \mathbf{a} = \frac{\Delta \mathbf{v}}{t} \]
Substituting the values:
\[ \mathbf{a} = \frac{-6\hat{j}}{2} = -3\hat{j} \, \text{m/s}^2 \]

### Final Answer
The acceleration of the particle is:
\[ \mathbf{a} = -3\hat{j} \, \text{m/s}^2 \]

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