The velocity vector of the motion described by the position vector of a particle, `r = 2t hati + t^(2) hatj` is given by

Options

A. `v = 2hati + 2thatj`
B. `v = 2t hat I + 2thatj`
C. `v = t hati +t^(2) hatj`
D. `v = 2hati +t^(2) hatj`

Explanation

To find the velocity vector of the motion described by the position vector \( \mathbf{r} = 2t \hat{i} + t^2 \hat{j} \), we will follow these steps:

### Step 1: Understand the Position Vector
The position vector of the particle is given as:
\[ \mathbf{r} = 2t \hat{i} + t^2 \hat{j} \]
Here, \( \hat{i} \) and \( \hat{j} \) are the unit vectors in the x and y directions, respectively.

### Step 2: Differentiate the Position Vector
The velocity vector \( \mathbf{v} \) is defined as the rate of change of the position vector with respect to time. Mathematically, this is expressed as:
\[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \]
We will differentiate each component of the position vector \( \mathbf{r} \) with respect to time \( t \).

### Step 3: Differentiate Each Component
1. Differentiate the \( x \)-component \( 2t \):
\[ \frac{d(2t)}{dt} = 2 \]
So, the \( x \)-component of the velocity vector is \( 2 \hat{i} \).

2. Differentiate the \( y \)-component \( t^2 \):
\[ \frac{d(t^2)}{dt} = 2t \]
So, the \( y \)-component of the velocity vector is \( 2t \hat{j} \).

### Step 4: Combine the Components
Now, we can combine the differentiated components to write the velocity vector:
\[ \mathbf{v} = 2 \hat{i} + 2t \hat{j} \]

### Step 5: Final Result
Thus, the velocity vector of the motion described by the position vector is:
\[ \mathbf{v} = 2 \hat{i} + 2t \hat{j} \]

### Conclusion
The correct answer is:
\[ \mathbf{v} = 2 \hat{i} + 2t \hat{j} \]