STATEMENT -1 : For an observer looking out through the window of a fast moving train , the nearby objects appear to move in the opposite direction to the train , while the distant objects appear to be stationary .
STATEMENT - 2 : If the observer and the object are moving at velocities ` vec v_(1)` and `vec v_(2)` respecttively with refrence to a laboratory frame , the velocity of the object with respect to a laboratory frame , the velocity of the object with respect to the observer is `vecv_(2) - vecv(1) ` .
(a) Statement -1 is True, statement -2 is true , statement -2 is a correct explanation for statement -1
(b) Statement 1 is True , Statement -2 is True , statement -2 is NOT a correct explanation for statement -1
(c) Statement - 1 is True , Statement -2 is False
(d) Statement -1 is False, Statement -2 is True

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-2
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we will analyze both statements step by step to determine their validity and the relationship between them.

### Step 1: Analyze Statement 1
**Statement 1:** For an observer looking out through the window of a fast-moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary.

- When a train moves quickly, the observer inside the train sees nearby objects (like trees, buildings) moving past them in the opposite direction to the train's motion. This is due to the relative motion between the observer and the nearby objects.
- Distant objects (like mountains or the moon) appear stationary because the angle of view changes very little over time, making their motion imperceptible to the observer.

**Conclusion for Statement 1:** True.

### Step 2: Analyze Statement 2
**Statement 2:** If the observer and the object are moving at velocities \( \vec{v_1} \) and \( \vec{v_2} \) respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is \( \vec{v_2} - \vec{v_1} \).

- This statement describes how to calculate the relative velocity of an object as seen by an observer. If the observer is moving at velocity \( \vec{v_1} \) and the object is moving at velocity \( \vec{v_2} \), then the relative velocity of the object with respect to the observer is indeed calculated as \( \vec{v_2} - \vec{v_1} \).
- This formula is a standard result in kinematics for finding relative velocities.

**Conclusion for Statement 2:** True.

### Step 3: Determine the Relationship Between the Statements
- Statement 2 provides a mathematical explanation for the observations made in Statement 1. The relative velocity formula helps to explain why nearby objects appear to move in the opposite direction while distant objects appear stationary.
- Since both statements are true and Statement 2 correctly explains Statement 1, we can conclude that Statement 2 is a correct explanation for Statement 1.

### Final Conclusion
Both statements are true, and Statement 2 is a correct explanation for Statement 1. Therefore, the correct option is:
**(a) Statement 1 is True, Statement 2 is True, Statement 2 is a correct explanation for Statement 1.**

---