The position vector of a particle is given by `vec r = (2t hati+5t^(2)hatj)m` (t is time in sec). Then the angle between initial velocity and initial acceleration is

Options

A. zero
B. `45^(@)`
C. `90^(@)`
D. `180^(@)`

Explanation

To find the angle between the initial velocity and initial acceleration of the particle whose position vector is given by \(\vec{r} = (2t \hat{i} + 5t^2 \hat{j})\) m, we can follow these steps:

### Step 1: Find the velocity vector
The velocity vector \(\vec{v}\) is the derivative of the position vector \(\vec{r}\) with respect to time \(t\).

\[ \vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(2t \hat{i} + 5t^2 \hat{j}) \]

Calculating the derivative:

\[ \vec{v} = 2 \hat{i} + 10t \hat{j} \]

### Step 2: Find the initial velocity
To find the initial velocity, we evaluate \(\vec{v}\) at \(t = 0\).

\[ \vec{v}(0) = 2 \hat{i} + 10(0) \hat{j} = 2 \hat{i} \]

So, the initial velocity vector is:

\[ \vec{v}_0 = 2 \hat{i} \]

### Step 3: Find the acceleration vector
The acceleration vector \(\vec{a}\) is the derivative of the velocity vector \(\vec{v}\) with respect to time \(t\).

\[ \vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(2 \hat{i} + 10t \hat{j}) \]

Calculating the derivative:

\[ \vec{a} = 0 \hat{i} + 10 \hat{j} = 10 \hat{j} \]

### Step 4: Find the initial acceleration
The initial acceleration vector is:

\[ \vec{a}_0 = 10 \hat{j} \]

### Step 5: Calculate the angle between the initial velocity and initial acceleration
To find the angle \(\theta\) between the vectors \(\vec{v}_0\) and \(\vec{a}_0\), we can use the dot product formula:

\[ \vec{v}_0 \cdot \vec{a}_0 = |\vec{v}_0| |\vec{a}_0| \cos(\theta) \]

Calculating the magnitudes:

\[ |\vec{v}_0| = \sqrt{(2)^2} = 2 \]
\[ |\vec{a}_0| = \sqrt{(10)^2} = 10 \]

Calculating the dot product:

\[ \vec{v}_0 \cdot \vec{a}_0 = (2 \hat{i}) \cdot (10 \hat{j}) = 0 \]

Since the dot product is zero, we can conclude that:

\[ \cos(\theta) = 0 \implies \theta = 90^\circ \]

### Final Answer
The angle between the initial velocity and initial acceleration is \(90^\circ\).

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