Relative Motion: Train Crossing a Man and a Platform

We analyze a classic relative motion problem: how long a moving train takes to completely pass a man and a platform.

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Given Data

• Length of train: $L_t = 120\,\text{m}$
• Speed of train: $v = 54\,\text{km/h}$
• Length of platform: $L_p = 180\,\text{m}$

First, convert the speed from km/h to m/s:

$$54\,\text{km/h} = 54 \times \frac{1000}{3600} = 54 \times \frac{5}{18} = 15\,\text{m/s}$$

So, $v = 15\,\text{m/s}$.

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Time to Cross a Man (Point Object)

To completely cross a man, the train’s whole length must pass him.

Distance to be covered relative to the man:

$$D_{\text{man}} = L_t = 120\,\text{m}$$

Using $t = \frac{\text{distance}}{\text{speed}}$:

$$t_{\text{man}} = \frac{D_{\text{man}}}{v} = \frac{120}{15} = 8\,\text{s}$$

Answer: It takes 8 seconds for the train to cross the man.

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Time to Cross a Platform

To completely cross the platform, the entire train must go from just before entering the platform to just after leaving it.

Total distance covered (train’s front moves from start of platform to its end, while the back has to clear the platform end):

$$D_{\text{platform}} = L_t + L_p = 120 + 180 = 300\,\text{m}$$

Time:

$$t_{\text{platform}} = \frac{D_{\text{platform}}}{v} = \frac{300}{15} = 20\,\text{s}$$

Answer: It takes 20 seconds for the train to cross the platform.

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What the Visualization Shows

This visualization animates:

• A cyan train moving along a track.
• A man (small marker) on the platform.
• A platform segment with adjustable length.
• You can change the train’s speed, length, and platform length to see how the crossing times change.

Two key formulas are highlighted:

• Crossing a man: $t = \dfrac{L_t}{v}$
• Crossing a platform: $t = \dfrac{L_t + L_p}{v}$

Times are displayed numerically and also encoded visually by the train’s progress over time.