When a particle is undergoing motion, the diplacement of the particle has a magnitude that is equal to or smaller than the total distance travelled by the particle. In many cases the displacement of the particle may actually be zero, while the distance travelled by it is non-zero. Both these quantities, however depend on the frame of reference in which motion of the particle is being observed. Consider a particle which is projected in the earth's gravitational field, close to its surface, with a speed of `100sqrt(2) m//s`, at an angle of `45^(@)` with the horizontal in the eastward direction. Ignore air resistance and assume that the acceleration due to gravity is `10 m//s^(2)`.
There exists a frame (D) in which the distance travelled by the particle is minimum. This minimum distance is equal to :-

Options

A. 2 km
B. 1 km
C. 0 km
D. 500 m

Explanation

To solve the problem, we need to analyze the motion of a particle projected in the Earth's gravitational field and determine the minimum distance traveled by the particle in a specific frame of reference.

### Step-by-step Solution:

1. **Understanding the Motion**:
The particle is projected with an initial speed of \(100\sqrt{2} \, \text{m/s}\) at an angle of \(45^\circ\) to the horizontal. This means that the particle will undergo projectile motion under the influence of gravity.

2. **Components of Initial Velocity**:
We can break down the initial velocity into its horizontal and vertical components:
\[ v_x = v \cos(45^\circ) = 100\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 100 \, \text{m/s} \]
\[ v_y = v \sin(45^\circ) = 100\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 100 \, \text{m/s} \]

3. **Time of Flight**:
The time of flight \(T\) for a projectile is given by:
\[ T = \frac{2v_y}{g} = \frac{2 \cdot 100}{10} = 20 \, \text{s} \]

4. **Range of the Projectile**:
The horizontal distance (range) traveled by the projectile can be calculated using:
\[ R = v_x \cdot T = 100 \cdot 20 = 2000 \, \text{m} \]

5. **Frame of Reference**:
The problem states that there exists a frame (D) in which the distance traveled by the particle is minimum. If we consider a frame of reference that moves with the particle, the particle will appear to be at rest in that frame.

6. **Minimum Distance in the Moving Frame**:
In the frame of reference that moves with the particle, the distance traveled by the particle is zero. This is because, from the perspective of this frame, the particle does not move; it remains at the same position relative to the observer in that frame.

7. **Conclusion**:
Therefore, the minimum distance traveled by the particle in the frame (D) where it is at rest is:
\[ \text{Minimum Distance} = 0 \, \text{m} \]

### Final Answer:
The minimum distance traveled by the particle in the frame (D) is \(0 \, \text{m}\).