An object starts from rest, and moves under the acceleration ` veca = 4hati` . Its position after 3 s is given by ` vecr = 7 hati + 4hatj` . What is its initial position ?

Answer

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

### Step 1: Understand the motion parameters
The object starts from rest, which means its initial velocity \( \vec{u} = 0 \). The acceleration is given as \( \vec{a} = 4 \hat{i} \) m/s².

### Step 2: Use the second equation of motion
The second equation of motion states that:
\[ \vec{r} = \vec{u} t + \frac{1}{2} \vec{a} t^2 \]
Since the initial velocity \( \vec{u} = 0 \), the equation simplifies to:
\[ \vec{r} = \frac{1}{2} \vec{a} t^2 \]

### Step 3: Substitute the values into the equation
Substituting the acceleration and time into the equation:
\[ \vec{r} = \frac{1}{2} (4 \hat{i}) (3^2) \]
Calculating \( 3^2 = 9 \):
\[ \vec{r} = \frac{1}{2} (4 \hat{i}) (9) = 2 \cdot 9 \hat{i} = 18 \hat{i} \]

### Step 4: Relate the position to the initial position
Let the initial position be \( \vec{r_0} = p \hat{i} + q \hat{j} \). After 3 seconds, the position is given as:
\[ \vec{r} = \vec{r_0} + 18 \hat{i} \]
Given that \( \vec{r} = 7 \hat{i} + 4 \hat{j} \), we can write:
\[ p + 18 = 7 \quad \text{(for the } \hat{i} \text{ component)} \]
\[ q = 4 \quad \text{(for the } \hat{j} \text{ component)} \]

### Step 5: Solve for \( p \) and \( q \)
From the equation \( p + 18 = 7 \):
\[ p = 7 - 18 = -11 \]
And from the \( \hat{j} \) component:
\[ q = 4 \]

### Step 6: Write the initial position
Thus, the initial position of the object is:
\[ \vec{r_0} = -11 \hat{i} + 4 \hat{j} \]

### Final Answer
The initial position of the object is:
\[ \vec{r_0} = -11 \hat{i} + 4 \hat{j} \]

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