Analysis of 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.

1. Nearby Objects: Let the velocity of the train (and the observer) with respect to the ground be $\vec{v}_{T}$. Let a nearby object (e.g., a tree) be stationary with respect to the ground, so its velocity is $\vec{v}_{obj} = 0$. The velocity of the object with respect to the observer is given by the relative velocity formula: $\vec{v}_{obj, obs} = \vec{v}_{obj} - \vec{v}_{T} = 0 - \vec{v}_{T} = -\vec{v}_{T}$ The negative sign indicates that the object appears to move in the direction opposite to the train's motion. This part of the statement is correct.

2. Distant Objects: The perception of motion depends not just on the linear relative velocity but on the angular velocity of the object with respect to the observer. The angular velocity, $\omega$, is the rate at which the line of sight to the object rotates. The relationship between linear velocity ($v_{\perp}$, the component perpendicular to the line of sight) and angular velocity is $\omega = \frac{v_{\perp}}{r}$, where $r$ is the distance to the object.

3. For both a nearby object and a distant object (like a far-off mountain), if they are stationary on the ground, their linear velocity relative to the observer is the same, i.e., $-\vec{v}_{T}$.

4. However, the distance $r$ to these objects is very different. Let $r_{near}$ be the distance to the nearby object and $r_{distant}$ be the distance to the distant object. We have $r_{distant} \gg r_{near}$.

5. The angular velocity of the nearby object is $\omega_{near} \approx \frac{|\vec{v}_{T}|}{r_{near}}$. The angular velocity of the distant object is $\omega_{distant} \approx \frac{|\vec{v}_{T}|}{r_{distant}}$.

6. Since $r_{distant}$ is very large, $\omega_{distant}$ is very small, approaching zero. Our eyes perceive this very small angular velocity as the object being stationary.

7. Therefore, Statement-1 is a correct description of the phenomenon of parallax and is factually true.

Analysis of Statement-2

STATEMENT-2: If the observer and the object are moving at velocities ${\overrightarrow v _1}$ and ${\overrightarrow v _2}$ respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is ${\overrightarrow v _2}$-${\overrightarrow v _1}$.

1. This is the fundamental definition of relative velocity in classical mechanics.
2. Let $\vec{r}_1$ be the position vector of the observer and $\vec{r}_2$ be the position vector of the object in a laboratory frame.
3. The position of the object with respect to the observer is $\vec{r}_{21} = \vec{r}_2 - \vec{r}_1$.
4. Differentiating with respect to time to find the relative velocity: $\vec{v}_{21} = \frac{d\vec{r}_{21}}{dt} = \frac{d\vec{r}_2}{dt} - \frac{d\vec{r}_1}{dt} = \vec{v}_2 - \vec{v}_1$
5. Thus, Statement-2 is a correct physical principle.

Relationship between Statement-1 and Statement-2

1. We have established that both Statement-1 and Statement-2 are true.
2. Now we must determine if Statement-2 is the correct explanation for Statement-1.
3. Statement-2 explains why nearby objects appear to move backward. Using the formula from Statement-2, the relative velocity of any stationary object (near or far) is $-\vec{v}_{T}$.
4. However, Statement-2 alone does not explain why distant objects appear stationary. According to Statement-2, a distant stationary mountain has the same linear relative velocity $(-\vec{v}_{T})$ as a nearby tree. This contradicts the observation that the mountain appears stationary.
5. The full explanation for Statement-1 requires not only the concept of relative linear velocity (from Statement-2) but also the concept of angular velocity and how it depends on distance ($r$). The apparent stationarity of distant objects is due to their very small angular velocity, which Statement-2 does not address.
6. Therefore, while Statement-2 is a true statement and is a part of the explanation, it is not the complete or correct explanation for the entirety of Statement-1.

Conclusion

• Statement-1 is True.
• Statement-2 is True.
• Statement-2 is NOT a correct explanation for Statement-1.

This corresponds to option B.