A particle travels with speed `100m//s` from the print `(10, 20)` in a direction `24hat(i)+7hat(j)`. Find its position verctor after `2` seconds.

Options

A. `202hat(i)+76hat(j)`
B. `204hat(i)+78hat(j)`
C. `40hat(i)-151hat(j)`
D. `45hat(i)-151hat(j)`

Explanation

To find the position vector of the particle after 2 seconds, we can follow these steps:

### Step 1: Determine the unit vector in the direction of motion
The direction of motion is given as \( 24\hat{i} + 7\hat{j} \). First, we need to find the magnitude of this vector.

\[ \text{Magnitude} = \sqrt{(24)^2 + (7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

### Step 2: Calculate the unit vector
Now, we can find the unit vector in the direction of motion:

\[ \text{Unit vector} = \frac{24\hat{i} + 7\hat{j}}{25} = \frac{24}{25}\hat{i} + \frac{7}{25}\hat{j} \]

### Step 3: Calculate the velocity vector
The speed of the particle is given as \( 100 \, \text{m/s} \). Therefore, the velocity vector \( \vec{v} \) can be calculated as:

\[ \vec{v} = 100 \left( \frac{24}{25}\hat{i} + \frac{7}{25}\hat{j} \right) = 96\hat{i} + 28\hat{j} \]

### Step 4: Calculate the displacement after 2 seconds
Displacement \( \vec{s} \) can be calculated using the formula:

\[ \vec{s} = \vec{v} \cdot t \]

Substituting the values:

\[ \vec{s} = (96\hat{i} + 28\hat{j}) \cdot 2 = 192\hat{i} + 56\hat{j} \]

### Step 5: Determine the initial position vector
The initial position vector \( \vec{r_0} \) is given as:

\[ \vec{r_0} = 10\hat{i} + 20\hat{j} \]

### Step 6: Calculate the final position vector
The final position vector \( \vec{r} \) after 2 seconds can be found by adding the initial position vector and the displacement:

\[ \vec{r} = \vec{r_0} + \vec{s} = (10\hat{i} + 20\hat{j}) + (192\hat{i} + 56\hat{j}) \]

Combining the components:

\[ \vec{r} = (10 + 192)\hat{i} + (20 + 56)\hat{j} = 202\hat{i} + 76\hat{j} \]

### Final Answer
The position vector after 2 seconds is:

\[ \vec{r} = 202\hat{i} + 76\hat{j} \]

---