Explain this common observation clearly: If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move rapidly in a direction opposite to the train’s motion, but the distant objects (hill tops, the Moon, the stars etc.) seem to be stationary. (In fact, since you are aware that you are moving, these distant objects seem to move with you).
As distance $r \to \infty$, angular speed $\omega \to 0$ (remains stationary)!
Looking Out The Express Train Window
Observe the multi-plane parallax: trackside poles flash past at high speed, hills glide slowly, and the Moon appears frozen in position, following along.
Notice how the telegraph poles flash backward so fast they blur, while the Moon stays fixed right in the center of your view the whole journey.
Top-Down Ray Diagram & Line of Sight Rotation
See the moving train from above. Watch how the line of sight (LOS) vector joining your eye to an object rotates through angle $\Delta\theta$.
Why Do Distant Objects Seem to Move WITH You?
The second half of the question addresses a fascinating trick of human perception: when you know you are moving forward, stationary objects can feel like they are moving alongside you.
Your eyes naturally measure relative motion against the ground immediately outside the train. Trees and ground rush backwards at $-\vec{v}_{\text{train}}$. In your subconscious frame of reference, "everything stationary is moving backward".
Because the Moon’s line of sight does not rotate backward ($\omega \approx 0$), it refuses to slip behind the window frame. Relative to the trees rushing backward, the Moon is advancing forward at $+\vec{v}$!
Since you are consciously aware that your train has covered 50 kilometers, yet the Moon is still in the exact same window pane right next to you, your visual cortex perceives that the Moon has travelled those 50 kilometers with you.
Drag the distance slider below to see how quickly the apparent angular rotation rate drops from whizzing speed down to absolute zero.
Formal Physics Explanation (Standard NCERT Format)
Board exam ready analytical answer with trigonometry, angular velocity formulas, and cognitive reasoning.
The apparent motion of any object depends upon the rate of change of angle of the line of sight joining the observer’s eye to that object.
Where $v$ is the train's linear velocity and $r$ is the perpendicular distance of the object from the train track.
For nearby trees, telephone poles, and houses:
The line of sight rotates through a large angle in a fraction of a second. Consequently, these objects appear to rapidly rush in a direction opposite to the train's motion.
For distant objects like hills, the Moon, or stars:
The line of sight remains virtually parallel during the journey. Because you are moving forward but the direction to the distant object never shifts, it appears to move with you!
The phenomenon is completely explained by motion parallax: the angular velocity of the line of sight is inversely proportional to the distance of the object ($\omega = \frac{v}{r}$).