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Trains

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Quick recall

  • Crossing a platform requires train length plus platform length.
  • Convert every speed to the same unit; use the positive relative speed.

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Formula Cards

1. Cross a pole or stationary person

t=Lvt=\frac{L}{v}

L = train length.

2. Cross a platform

t=L+Pvt=\frac{L+P}{v}

P = platform length.

3. Two trains, opposite directions

t=L1+L2v1+v2t=\frac{L_1+L_2}{v_1+v_2}

4. Two trains, same direction

t=L1+L2∣v1−v2∣t=\frac{L_1+L_2}{|v_1-v_2|}

Speeds must differ for one train to overtake the other.

5. Moving person

t=Lvrelativet=\frac{L}{v_{\text{relative}}}

Treat the person as a point. Add speeds if opposite; take their positive difference if in the same direction.

Worked Examples

Example 1

Question: A 200 m train travels at 20 m/s. Find its time to cross a pole.

Step 1: Use the train length as the crossing distance.

D=200 mD=200\text{ m}

Step 2: Divide by speed.

t=20020=10 st=\frac{200}{20}=10\text{ s}

Answer: 10 seconds.

Example 2

Question: A 150 m train crosses a 250 m platform at 20 m/s. Find the time.

Step 1: Add train and platform lengths.

D=150+250=400 mD=150+250=400\text{ m}

Step 2: Divide by speed.

t=40020=20 st=\frac{400}{20}=20\text{ s}

Answer: 20 seconds.

Example 3

Question: Trains of 150 m and 100 m approach at 30 m/s and 20 m/s. Find crossing time.

Step 1: Add lengths and approaching speeds.

D=250 m,vrelative=50 m/sD=250\text{ m},\qquad v_{\text{relative}}=50\text{ m/s}

Step 2: Divide.

t=25050=5 st=\frac{250}{50}=5\text{ s}

Answer: 5 seconds.

Example 4

Question: The same trains travel in the same direction at 30 m/s and 20 m/s. Find overtaking time.

Step 1: Subtract speeds; add lengths.

vrelative=30−20=10 m/s,D=250 mv_{\text{relative}}=30-20=10\text{ m/s},\qquad D=250\text{ m}

Step 2: Divide.

t=25010=25 st=\frac{250}{10}=25\text{ s}

Answer: 25 seconds.

Drill

Try each question before opening its answer.

Question 1

Train 120m crosses a pole in 8s. Find speed.

  • Show answer

    15 m/s

Question 2

Train 180m, speed 54km/h, crosses a platform in 20s. Find platform length.

  • Show answer

    120m

Question 3

Two trains 200m and 300m cross each other (opposite directions) in 10s at combined speed. Find combined speed.

  • Show answer

    50 m/s

Question 4

A train crosses a man standing on a platform in 9s and crosses the 240m platform in 24s. Find train's length and speed.

  • Show answer

    144m, 16 m/s

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