A particle's velocity changes from `(2hat I +3 hatj) ms^(-1)` in to `(3 hati - 2hatj) ms^(-1)` in 2s. Its average acceleration in `ms^(-2)` is

Options

A. `-(hati + 5hatj)`
B. `(hati +5 hatj)//2`
C. zero
D. `(hati -5hatj)//2`

Explanation

To find the average acceleration of the particle, we will 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:
\[ \mathbf{v} = 3 \hat{i} - 2 \hat{j} \, \text{m/s} \]

### Step 2: Calculate the change in velocity
The change in velocity \( \Delta \mathbf{v} \) is calculated as:
\[ \Delta \mathbf{v} = \mathbf{v} - \mathbf{u} \]
Substituting the values:
\[ \Delta \mathbf{v} = (3 \hat{i} - 2 \hat{j}) - (2 \hat{i} + 3 \hat{j}) \]
This simplifies to:
\[ \Delta \mathbf{v} = (3 \hat{i} - 2 \hat{j} - 2 \hat{i} - 3 \hat{j}) = (3 - 2) \hat{i} + (-2 - 3) \hat{j} = 1 \hat{i} - 5 \hat{j} \]

### Step 3: Calculate the average acceleration
The average acceleration \( \mathbf{a}_{avg} \) is given by the formula:
\[ \mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t} \]
Where \( \Delta t \) is the change in time, which is given as 2 seconds. Thus:
\[ \mathbf{a}_{avg} = \frac{1 \hat{i} - 5 \hat{j}}{2} \]
This simplifies to:
\[ \mathbf{a}_{avg} = \frac{1}{2} \hat{i} - \frac{5}{2} \hat{j} \, \text{m/s}^2 \]

### Final Result
The average acceleration of the particle is:
\[ \mathbf{a}_{avg} = 0.5 \hat{i} - 2.5 \hat{j} \, \text{m/s}^2 \]

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