The position of a particle x (in meters) at a time t seconds is given by the relation `vecr = (3t hati - t^2 hatj + 4 hatk)`. Calculate the magnitude of velocity of the particle after 5 seconds

Options

A. 3.55
B. 5.03
C. 8.75
D. 10.44

Explanation

To solve the problem, we need to calculate the magnitude of the velocity of the particle after 5 seconds, given the position vector \( \vec{r} = (3t \hat{i} - t^2 \hat{j} + 4 \hat{k}) \).

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

\[ \vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(3t \hat{i} - t^2 \hat{j} + 4 \hat{k}) \]

### Step 2: Differentiate each component
Now we differentiate each component of the position vector:

1. The \( \hat{i} \) component:
\[ \frac{d}{dt}(3t) = 3 \]

2. The \( \hat{j} \) component:
\[ \frac{d}{dt}(-t^2) = -2t \]

3. The \( \hat{k} \) component:
\[ \frac{d}{dt}(4) = 0 \]

Combining these results, we get the velocity vector:
\[ \vec{v} = 3 \hat{i} - 2t \hat{j} + 0 \hat{k} \]

### Step 3: Substitute \( t = 5 \) seconds into the velocity vector
Now we substitute \( t = 5 \) seconds into the velocity vector to find the velocity at that time:
\[ \vec{v}(5) = 3 \hat{i} - 2(5) \hat{j} + 0 \hat{k} \]
\[ \vec{v}(5) = 3 \hat{i} - 10 \hat{j} \]

### Step 4: Calculate the magnitude of the velocity vector
The magnitude of the velocity vector \( \vec{v} \) is given by:
\[ |\vec{v}| = \sqrt{(v_x)^2 + (v_y)^2 + (v_z)^2} \]
Substituting the components:
\[ |\vec{v}| = \sqrt{(3)^2 + (-10)^2 + (0)^2} \]
\[ |\vec{v}| = \sqrt{9 + 100 + 0} = \sqrt{109} \]

### Step 5: Calculate the numerical value
Now we calculate \( \sqrt{109} \):
\[ |\vec{v}| \approx 10.44 \text{ m/s} \]

### Final Answer
The magnitude of the velocity of the particle after 5 seconds is approximately \( 10.44 \text{ m/s} \).

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