Train, Bogies, and Passing a Pole – Proportional Reasoning

We are given:

• A train with 12 bogies (coaches) takes 24 seconds to completely pass a pole.
• At every station, 2 bogies are removed.
• After stopping at 3 stations, the train again passes a pole at the same speed.

We are asked:

How much time will the train now take to completely pass the pole?

---

Step 1: Understand the geometry of the situation

To completely pass a pole, the entire length of the train must go past the pole. So the time taken depends on:

• the length of the train, call it $L$
• the speed of the train, call it $v$

The relationship is:

$$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{L}{v}$$

Initially:

• Number of bogies: $12$
• Let the length of one bogie be $\ell$.
• Then total length: $L_{\text{initial}} = 12\ell$.
• Time taken to pass the pole: $T_{\text{initial}} = 24\,\text{s}$.

So, using

$$T_{\text{initial}} = \frac{L_{\text{initial}}}{v} = \frac{12\ell}{v} = 24$$

We don’t actually need $\ell$ or $v$ individually; we just use proportionality.

---

Step 2: After 3 stations – new number of bogies

At each station, 2 bogies are removed.

• After station 1: $12 - 2 = 10$ bogies
• After station 2: $10 - 2 = 8$ bogies
• After station 3: $8 - 2 = 6$ bogies

So the new train has 6 bogies.

Thus the new train length is

$$L_{\text{new}} = 6\ell$$

We are told the speed is unchanged:

$$v_{\text{new}} = v_{\text{initial}} = v$$

---

Step 3: Use proportional reasoning

Time to pass the pole is length divided by speed. Since the speed is constant, the time is directly proportional to the train’s length.

$$\frac{T_{\text{new}}}{T_{\text{initial}}} = \frac{L_{\text{new}}}{L_{\text{initial}}}$$

Substitute:

• $L_{\text{initial}} = 12\ell$
• $L_{\text{new}} = 6\ell$
• $T_{\text{initial}} = 24\,\text{s}$

$$\frac{T_{\text{new}}}{24} = \frac{6\ell}{12\ell} = \frac{6}{12} = \frac{1}{2}$$

So:

$$T_{\text{new}} = 24 \times \frac{1}{2} = 12\,\text{s}$$

---

Final Answer

After removing bogies at 3 stations (ending with 6 bogies) and keeping the same speed, the train will take

$$\boxed{12\ \text{seconds}}$$

to completely pass the pole.

---

About the Visualization

This interactive visualization shows:

• A side-view of the train as colored rectangles (bogies) moving past a vertical pole.
• You can adjust the number of bogies, the speed, and see how the time to pass the pole changes.
• A "shrink factor" slider smoothly morphs the train from 12 bogies down to 6 bogies to visually encode the idea that time is proportional to length when speed is fixed.

Key mathematical idea visualized:

$$T \propto L \quad \text{when} \quad v = \text{constant}$$

The overlay text on the canvas displays:

• Initial bogies and time (12 bogies, 24 s)
• Current bogies and proportional time prediction (e.g., 6 bogies → 12 s)

Use the animation to build intuition for why halving the number of bogies halves the time to pass the pole, when the train runs at the same speed.