Velocity of a particle at some instant is `v=(3hat i + 4hat j + 5hat k) m//s`. Find speed of the particle at this instant.

Options

A. 12
B. 7
C. -12
D. `5sqrt2`

Explanation

To find the speed of a particle given its velocity vector, we can follow these steps:

### Step 1: Identify the velocity vector
The velocity vector is given as:
\[ \mathbf{v} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \, \text{m/s} \]

### Step 2: Understand the relationship between speed and velocity
Speed is the magnitude of the velocity vector. Therefore, we need to calculate the magnitude of the velocity vector.

### Step 3: Calculate the magnitude of the velocity vector
The magnitude of a vector \(\mathbf{v} = a \hat{i} + b \hat{j} + c \hat{k}\) is given by the formula:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
In our case, \(a = 3\), \(b = 4\), and \(c = 5\).

### Step 4: Substitute the values into the formula
Substituting the values into the formula, we get:
\[ |\mathbf{v}| = \sqrt{3^2 + 4^2 + 5^2} \]

### Step 5: Calculate the squares
Calculating the squares:
\[ 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25 \]

### Step 6: Sum the squares
Now, sum these values:
\[ 9 + 16 + 25 = 50 \]

### Step 7: Take the square root
Now, take the square root of the sum:
\[ |\mathbf{v}| = \sqrt{50} \]

### Step 8: Simplify the square root
The square root of 50 can be simplified:
\[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \]

### Step 9: State the final answer
Thus, the speed of the particle at this instant is:
\[ \text{Speed} = 5\sqrt{2} \, \text{m/s} \]