The position vector of an object at any time t is given by `3t^(2) hati +6t hayj +hatk`. Its velocity along y-axis has the magnitude

Options

A. 6t
B. 6
C. 0
D. 9

Explanation

To find the magnitude of the velocity of the object along the y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Position Vector**:
The position vector \(\mathbf{r}(t)\) is given as:
\[ \mathbf{r}(t) = 3t^2 \hat{i} + 6t \hat{j} + \hat{k} \]

2. **Differentiate the Position Vector**:
To find the velocity vector \(\mathbf{v}(t)\), we need to differentiate the position vector with respect to time \(t\):
\[ \mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \frac{d}{dt}(3t^2 \hat{i} + 6t \hat{j} + \hat{k}) \]

3. **Calculate the Derivative**:
Differentiate each component:
\[ \mathbf{v}(t) = \frac{d}{dt}(3t^2) \hat{i} + \frac{d}{dt}(6t) \hat{j} + \frac{d}{dt}(1) \hat{k} \]
This results in:
\[ \mathbf{v}(t) = (6t) \hat{i} + (6) \hat{j} + (0) \hat{k} \]
Therefore, the velocity vector is:
\[ \mathbf{v}(t) = 6t \hat{i} + 6 \hat{j} \]

4. **Extract the y-component of Velocity**:
The y-component of the velocity vector is:
\[ v_y = 6 \text{ (constant)} \]

5. **Calculate the Magnitude of the Velocity along the y-axis**:
Since the y-component of the velocity is constant, the magnitude of the velocity along the y-axis is simply:
\[ |v_y| = 6 \]

### Final Answer:
The magnitude of the velocity along the y-axis is \(6\).

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