Time Calculation for Trains Crossing Platform

❓ Concept Question

How do we solve train crossing problems using relative speed?

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🖼 Concept Image

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✍️ Short Concept

This concept is based on:

👉 Relative speed
👉 Train crossing concept
👉 Time-distance relation.

Main idea:

$$\text{Time}= \frac{\text{Total Distance}}{\text{Relative Speed}}$$

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🔷 Step 1 — Relative Speed Formula 💯

For train crossing problems:

$$t=\frac{d}{v_{rel}}$$

Where:

• $t$ = time taken
• $d$ = total distance
• $v_{rel}$ = relative speed

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🔷 Step 2 — Same Direction Case

When two objects move in same direction:

$$v_{rel}=v_{fast}-v_{slow}$$

Example:

$$80-30=50\text{ km/h}$$

Convert into m/s:

$$50\times\frac{5}{18}$$
$$=13.89\text{ m/s}$$

If total length:

$$60+120=180\text{ m}$$

Then:

$$t=\frac{180}{13.89}$$
$$\approx12.96\text{ s}$$

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🔷 Step 3 — Opposite Direction Case

When objects move opposite to each other:

$$v_{rel}=v_1+v_2$$

Example:

$$80+30=110\text{ km/h}$$

Convert into m/s:

$$110\times\frac{5}{18}$$
$$=30.56\text{ m/s}$$

Now:

$$t=\frac{180}{30.56}$$
$$\approx5.89\text{ s}$$

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🔷 Step 4 — Key JEE Shortcut

Same direction:

$$\boxed{ v_{rel}=v_1-v_2 }$$

Opposite direction:

$$\boxed{ v_{rel}=v_1+v_2 }$$

Also:

$$\boxed{ t\propto\frac1{v_{rel}} }$$

Higher relative speed means lower crossing time.

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Same direction me speeds add kar dena

❌ km/h to m/s conversion bhool jaana

❌ Total distance me both lengths add na karna

Remember:

$$1\text{ km/h}=\frac{5}{18}\text{ m/s}$$

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✅ Final Takeaway

For train crossing:

$$\boxed{ t=\frac{\text{Total Length}}{\text{Relative Speed}} }$$

Always:

✔ Add speeds in opposite direction

✔ Subtract speeds in same direction

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⭐ Golden JEE Insight

If relative speed increases:

$$\text{Crossing time decreases}$$

because:

$$t\propto\frac1{v_{rel}}$$

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📚 Related Topics

• Direction of Motion from Velocity Vector at Given Time
• Find Diameter of Rotating Disc Using Torque and RPM
• Total Distance with Acceleration and Deceleration Phases