### Question 2.18

NCERT Class XI Physics

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 by Conceptual discussion

Nearer objects subtend greater angle at the eye. The change in the angle is greater so the near objects seem to move faster.

There is very less change in the angle of far away objects. So they appear stationary, and rather move along with us.

### Video Explanation

Please watch this youtube video for a quick explanation of the solution:

### Solution by Calculation

$ \begin{aligned} &\text{let distance of near object } = r \text{ meters}\\ &\text{let distance of far object } = R \text{ meters}\\\\ &\text{let } r \ll R\\\\ &\text{let speed of train } = +v \text{ m/s}\\\\ &\text{we can apply galilean transformations}\\ &\text{linear speed of near object} = -v \text{ m/s}\\ &\text{linear speed of far object is also} = -v \text{ m/s}\\\\ &\text{angular speed of near object }\omega_r = -\frac vr\\\\ &\text{angular speed of far object }\omega_R = -\frac vR\\\\ &\text{but } r \ll R \implies \omega_r \gg \omega_R\\\\ &\text{eye perceives angular change}\\ &\text{ so near objects are faster}\:\underline{Ans (1)}\\\\ &\underline{\underline \text {distant objects stationary}}\\ &\text{angular speed of far object }\omega_R = -\frac vR\\\\ &\text{but numerically } R \gg v \text{, so } \therefore \omega_R \to 0\\ &\therefore \text{far objects appear stationary} \:\underline{Ans (2)} \end{aligned} $

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