Step 1: Understanding the Concept:
The change in kinetic energy is given by the difference between the final kinetic energy and the initial kinetic energy. The kinetic energy of a body is determined by its mass and the square of its speed (magnitude of velocity).
Step 2: Key Formula or Approach:
1. Kinetic Energy (KE) is given by $KE = \frac{1}{2}mv^2$, where $v$ is the speed.
2. The change in kinetic energy is $\Delta KE = KE_{final} - KE_{initial} = \frac{1}{2}mv_{final}^2 - \frac{1}{2}mv_{initial}^2$.
3. For a velocity vector $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$, the square of the speed is $v^2 = |\vec{v}|^2 = v_x^2 + v_y^2 + v_z^2$.
Step 3: Detailed Explanation:
Given:
- Mass, $m = 2$ Kg.
- Initial velocity, $\vec{v}_{initial} = 3\hat{i} - 4\hat{j}$ m/s.
- Final velocity, $\vec{v}_{final} = 6\hat{j} + 2\hat{k}$ m/s.
First, calculate the square of the initial speed:
\[ v_{initial}^2 = |\vec{v}_{initial}|^2 = (3)^2 + (-4)^2 + (0)^2 = 9 + 16 = 25 \, (\text{m/s})^2 \]
Calculate the initial kinetic energy:
\[ KE_{initial} = \frac{1}{2}mv_{initial}^2 = \frac{1}{2}(2)(25) = 25 \text{ J} \]
Next, calculate the square of the final speed:
\[ v_{final}^2 = |\vec{v}_{final}|^2 = (0)^2 + (6)^2 + (2)^2 = 36 + 4 = 40 \, (\text{m/s})^2 \]
Calculate the final kinetic energy:
\[ KE_{final} = \frac{1}{2}mv_{final}^2 = \frac{1}{2}(2)(40) = 40 \text{ J} \]
Finally, calculate the change in kinetic energy:
\[ \Delta KE = KE_{final} - KE_{initial} = 40 \text{ J} - 25 \text{ J} = 15 \text{ J} \]
Step 4: Final Answer:
The change in kinetic energy of the body is 15 J. Therefore, option (A) is correct.