Apparent Motion of Nearby and Distant Objects from Moving Train

To determine the correct choice, let us analyze both statements individually:

1. Analysis of Statement-1:
Consider a train moving with velocity $\vec{V}_{\text{train}} = V\hat{i}$ relative to the ground. For any stationary object on the ground ($\vec{V}_{\text{object}} = \vec{0}$), its relative velocity with respect to the observer inside the train is:
$\vec{V}_{\text{rel}} = \vec{V}_{\text{object}} - \vec{V}_{\text{train}} = -V\hat{i}$

Thus, both nearby and distant objects have the same relative velocity vector relative to the observer. However, the visual perception of motion by the human eye depends on the rate of change of the angle of sight, which is quantified by the angular velocity $\omega$:
$\omega = \frac{V_{\perp}}{r}$
where $r$ is the distance from the observer to the object and $V_{\perp}$ is the component of relative velocity perpendicular to the line of sight.

• For nearby objects, the distance $r$ is small, resulting in a large angular velocity $\omega$. Hence, nearby objects appear to move rapidly past the observer in the opposite direction.
• For distant objects, the distance $r$ is extremely large, making the angular velocity $\omega \approx 0$. Consequently, distant objects appear to be practically stationary.

Therefore, Statement-1 is True.

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2. Analysis of Statement-2:
By the fundamental definition of relative velocity in classical kinematics, if an observer moves with velocity $\vec{V}_1$ and an object moves with velocity $\vec{V}_2$ relative to a reference (laboratory) frame, the velocity of the object relative to the observer is given by:
$\vec{V}_{\text{relative}} = \vec{V}_2 - \vec{V}_1$

Therefore, Statement-2 is True.

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3. Relationship between Statement-1 and Statement-2:
Statement-2 states that the relative linear velocity of all stationary objects ($\vec{V}_2 = \vec{0}$) with respect to the moving observer ($\vec{V}_1 = \vec{V}_{\text{train}}$) is identical, namely $-\vec{V}_1$, irrespective of their distance from the observer.

Because Statement-2 gives the same linear relative velocity for both nearby and distant objects, it fails to explain why distant objects appear stationary while nearby objects appear to move fast. The actual explanation lies in the difference in angular velocity ($\omega = V_{\perp}/r$) subtended at the observer's eye due to differing distances $r$.

Hence, Statement-2 is NOT the correct explanation for Statement-1.

Correct Answer: B (STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1)