The position vector of an object at any time t is given by `3t^(2) hati +6t hatj +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 along the y-axis for the given position vector, we will follow these steps:

### Step 1: Write down the position vector
The position vector of the object at any time \( t \) is given by:
\[ \vec{r}(t) = 3t^2 \hat{i} + 6t \hat{j} + \hat{k} \]

### Step 2: Differentiate the position vector to find the velocity vector
The velocity vector \( \vec{v}(t) \) is the derivative of the position vector with respect to time \( t \):
\[ \vec{v}(t) = \frac{d\vec{r}}{dt} \]
Differentiating each component:
- The \( \hat{i} \) component:
\[ \frac{d}{dt}(3t^2) = 6t \hat{i} \]
- The \( \hat{j} \) component:
\[ \frac{d}{dt}(6t) = 6 \hat{j} \]
- The \( \hat{k} \) component:
\[ \frac{d}{dt}(1) = 0 \hat{k} \]

Thus, the velocity vector is:
\[ \vec{v}(t) = 6t \hat{i} + 6 \hat{j} + 0 \hat{k} \]

### Step 3: Identify the y-component of the velocity
From the velocity vector, the y-component of the velocity is:
\[ v_y = 6 \hat{j} \]

### Step 4: Find the magnitude of the velocity along the y-axis
The magnitude of the velocity along the y-axis is simply the coefficient of \( \hat{j} \):
\[ |v_y| = 6 \text{ m/s} \]

### Conclusion
The magnitude of the velocity along the y-axis is:
\[ \boxed{6 \text{ m/s}} \]

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