If `v=(3hati+2hatj+6k)` m/s and `m=(2)/(7)kg` then find kinetic energy `(i.e. (1)/(2)mv^(2))`

Answer

To find the kinetic energy of the object given its velocity and mass, we can follow these steps:

### Step 1: Identify the given values
We have:
- Velocity vector \( \mathbf{v} = 3\hat{i} + 2\hat{j} + 6\hat{k} \) m/s
- Mass \( m = \frac{2}{7} \) kg

### Step 2: Calculate the magnitude of the velocity
The magnitude of the velocity \( v \) can be calculated using the formula:
\[ v = \sqrt{(v_x)^2 + (v_y)^2 + (v_z)^2} \]
where \( v_x = 3 \), \( v_y = 2 \), and \( v_z = 6 \).

Calculating:
\[ v = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \text{ m/s} \]

### Step 3: Use the kinetic energy formula
The formula for kinetic energy \( KE \) is given by:
\[ KE = \frac{1}{2} m v^2 \]

### Step 4: Substitute the values into the kinetic energy formula
Now substituting the mass and the calculated speed into the formula:
\[ KE = \frac{1}{2} \left(\frac{2}{7}\right) (7^2) \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
So,
\[ KE = \frac{1}{2} \left(\frac{2}{7}\right) (49) \]

### Step 5: Simplify the expression
Now, simplifying:
\[ KE = \frac{1}{2} \cdot \frac{2 \cdot 49}{7} = \frac{98}{14} = 7 \text{ Joules} \]

### Final Answer
The kinetic energy of the object is \( 7 \) Joules.

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