The position of an object changes from ` vecr = ( 2hati + hatj) " m to " vecr_(1) = ( 4hati + 3hati)` m in 2s. Find its average velocity.

Answer

To solve the problem of finding the average velocity of an object that changes its position from \(\vec{r} = (2\hat{i} + \hat{j}) \, \text{m}\) to \(\vec{r}_1 = (4\hat{i} + 3\hat{j}) \, \text{m}\) in 2 seconds, we can follow these steps:

### Step 1: Identify the initial and final positions
- Initial position: \(\vec{r} = 2\hat{i} + \hat{j}\)
- Final position: \(\vec{r}_1 = 4\hat{i} + 3\hat{j}\)

### Step 2: Calculate the displacement
Displacement \(\vec{s}\) is given by the difference between the final and initial positions:
\[ \vec{s} = \vec{r}_1 - \vec{r} = (4\hat{i} + 3\hat{j}) - (2\hat{i} + \hat{j}) \]
\[ \vec{s} = (4 - 2)\hat{i} + (3 - 1)\hat{j} = 2\hat{i} + 2\hat{j} \]

### Step 3: Calculate the magnitude of the displacement
The magnitude of the displacement \(|\vec{s}|\) can be calculated using the formula:
\[ |\vec{s}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \(x_1 = 2\), \(y_1 = 1\), \(x_2 = 4\), \(y_2 = 3\):
\[ |\vec{s}| = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{m} \]

### Step 4: Calculate the average velocity
Average velocity \(\vec{v}_{avg}\) is given by:
\[ \vec{v}_{avg} = \frac{\vec{s}}{t} \]
Where \(t = 2 \, \text{s}\):
\[ \vec{v}_{avg} = \frac{2\sqrt{2} \, \text{m}}{2 \, \text{s}} = \sqrt{2} \, \text{m/s} \]

### Final Answer
The average velocity of the object is \(\sqrt{2} \, \text{m/s}\).

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