The position vector of a particle is determined by the expression `vec r = 3t^2 hat i+ 4t^2 hat j + 7 hat k`. The displacement travelled in first `10` seconds is :

Options

A. 100 m
B. 150 m
C. 300 m
D. 500 m

Explanation

To solve the problem, we need to find the displacement of the particle over the first 10 seconds given its position vector. The position vector is given by:

\[ \vec{r} = 3t^2 \hat{i} + 4t^2 \hat{j} + 7 \hat{k} \]

### Step 1: Calculate the position vector at \( t = 0 \)

At \( t = 0 \):

\[ \vec{r_1} = 3(0)^2 \hat{i} + 4(0)^2 \hat{j} + 7 \hat{k} = 0 \hat{i} + 0 \hat{j} + 7 \hat{k} = 7 \hat{k} \]

### Step 2: Calculate the position vector at \( t = 10 \)

At \( t = 10 \):

\[ \vec{r_2} = 3(10)^2 \hat{i} + 4(10)^2 \hat{j} + 7 \hat{k} \]
\[ = 3(100) \hat{i} + 4(100) \hat{j} + 7 \hat{k} \]
\[ = 300 \hat{i} + 400 \hat{j} + 7 \hat{k} \]

### Step 3: Calculate the displacement

The displacement \( \vec{d} \) is given by:

\[ \vec{d} = \vec{r_2} - \vec{r_1} \]
\[ = (300 \hat{i} + 400 \hat{j} + 7 \hat{k}) - (0 \hat{i} + 0 \hat{j} + 7 \hat{k}) \]
\[ = 300 \hat{i} + 400 \hat{j} + 7 \hat{k} - 7 \hat{k} \]
\[ = 300 \hat{i} + 400 \hat{j} \]

### Step 4: Calculate the magnitude of the displacement

The magnitude of the displacement \( |\vec{d}| \) is calculated using the formula:

\[ |\vec{d}| = \sqrt{(300)^2 + (400)^2 + (0)^2} \]
\[ = \sqrt{90000 + 160000} \]
\[ = \sqrt{250000} \]
\[ = 500 \text{ meters} \]

### Final Answer

The displacement travelled in the first 10 seconds is **500 meters**.

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