The initial position of an object at rest is given by `3 hati - 8 hatj`. It moves with constant acceleration and reaches to the position `2 hati +4hatj` after 4s. What is its acceleration ?

Options

A. `-(1)/(8) hati + (3)/(2) hatj`
B. `2 hati - (1)/(8) hatj`
C. `-(1)/(2) hati + 8 hatj`
D. `8 hati -(3)/(2) hatj`

Explanation

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

### Step 1: Identify the initial and final position vectors
The initial position vector \( \mathbf{r_1} \) is given as:
\[ \mathbf{r_1} = 3 \hat{i} - 8 \hat{j} \]
The final position vector \( \mathbf{r_2} \) after 4 seconds is given as:
\[ \mathbf{r_2} = 2 \hat{i} + 4 \hat{j} \]

### Step 2: Calculate the displacement vector
The displacement vector \( \mathbf{s} \) can be calculated using the formula:
\[ \mathbf{s} = \mathbf{r_2} - \mathbf{r_1} \]
Substituting the values, we get:
\[ \mathbf{s} = (2 \hat{i} + 4 \hat{j}) - (3 \hat{i} - 8 \hat{j}) = 2 \hat{i} + 4 \hat{j} - 3 \hat{i} + 8 \hat{j} = -1 \hat{i} + 12 \hat{j} \]

### Step 3: Use the kinematic equation
Since the object is at rest initially, the initial velocity \( \mathbf{u} \) is:
\[ \mathbf{u} = 0 \hat{i} + 0 \hat{j} \]
We can use the kinematic equation for displacement:
\[ \mathbf{s} = \mathbf{u} t + \frac{1}{2} \mathbf{a} t^2 \]
Substituting the known values, we have:
\[ \mathbf{s} = 0 + \frac{1}{2} \mathbf{a} (4^2) \]
This simplifies to:
\[ \mathbf{s} = 8 \mathbf{a} \]

### Step 4: Set up the equation for acceleration
Now we can equate the displacement vector \( \mathbf{s} \) to \( 8 \mathbf{a} \):
\[ -1 \hat{i} + 12 \hat{j} = 8 \mathbf{a} \]
From this, we can solve for \( \mathbf{a} \):
\[ \mathbf{a} = \frac{-1 \hat{i} + 12 \hat{j}}{8} = -\frac{1}{8} \hat{i} + \frac{12}{8} \hat{j} = -\frac{1}{8} \hat{i} + \frac{3}{2} \hat{j} \]

### Step 5: Write the final answer
Thus, the acceleration vector \( \mathbf{a} \) is:
\[ \mathbf{a} = -\frac{1}{8} \hat{i} + \frac{3}{2} \hat{j} \]

### Conclusion
The correct answer is:
\[ \mathbf{a} = -\frac{1}{8} \hat{i} + \frac{3}{2} \hat{j} \]