Class 11 NCERT PHYSICS • Chapter 1: Units and Measurement • Question 16

QUESTION 16

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).

SOLUTION

We need to explain why nearby objects appear to move faster than distant ones when viewed from a moving train.

Concept: Parallax

The phenomenon is explained by parallax and angular velocity. The human eye perceives motion based on the rate at which the line of sight to an object changes, which is inversely proportional to the distance from the observer.

Given:

Velocity of the train: $v$
Distance to a nearby object: $r_{\text{near}}$
Distance to a distant object: $r_{\text{distant}}$
Condition: $r_{\text{distant}} \gg r_{\text{near}}$

Solving:

In the frame of reference of the train, the observer is at rest, and all stationary objects on the ground appear to move in the opposite direction with velocity $v$ .
The perceived speed of an object is determined by its angular velocity ( $\omega$ ), which is the rate of change of the angle ( $\theta$ ) of the line of sight. For an object at a perpendicular distance $r$ from the observer's path:

\omega = \frac{v}{r}

For nearby objects, such as trees or houses, the distance $r_{\text{near}}$ is small. Therefore:

\omega_{\text{near}} = \frac{v}{r_{\text{near}}}

Since $r_{\text{near}}$ is small, $\omega_{\text{near}}$ is large. The eye detects a rapid change in the angle of sight, causing the object to appear to move quickly in the opposite direction.

For distant objects, such as hilltops or the Moon, the distance $r_{\text{distant}}$ is very large. Therefore:

\omega_{\text{distant}} = \frac{v}{r_{\text{distant}}}

As $r_{\text{distant}} \to \infty$ , the angular velocity $\omega_{\text{distant}} \to 0$ . Because the change in the angle of sight is negligible over time, the object appears stationary.

The illusion of "moving with you" occurs because the brain knows the train is moving forward. Since the distant object's relative position (angle) does not change, the brain interprets this as the object maintaining its position relative to the train, thus appearing to move along with it.

Answer

The observation is due to the angular velocity being inversely proportional to distance:

\omega = \frac{v}{r}

Nearby objects have a high angular velocity and appear to move rapidly, while distant objects have a negligible angular velocity and appear stationary or move with the observer.