Two Trains Overtaking Each Other with Time Calculations
Two Trains Overtaking Each Other problems use relative speed and the combined length of both trains. When two trains move in the same direction, their relative speed is the difference of their speeds. The time taken to overtake depends on the sum of their lengths divided by this relative speed.
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What Happens When Two Trains Overtake Each Other?
If the train lengths are L₁ and L₂, and their speeds are S₁ and S₂, then the overtaking distance is L₁ + L₂. For trains moving in the same direction, relative speed is |S₁ − S₂|. Therefore, time taken is (L₁ + L₂) ÷ |S₁ − S₂|, provided the lengths and speeds use compatible units. For example, trains of lengths 120 m and 180 m moving at 72 km/h and 54 km/h take 300 ÷ 5 = 60 seconds to overtake completely, because their relative speed is 18 km/h = 5 m/s.
Two Trains Overtaking Each Other Video Lessons
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Trains Moving in the Same Direction
Learn how to solve train problems involving two trains moving in the same direction, including overtaking situations using relative speed and distance concepts.
Two Trains Overtaking Each Other Formula & Tricks
Important Formulas
The faster train must cover the length of the slower train as well as its own length.
Subtract the slower speed from the faster speed when both trains move in the same direction.
Use this when both train lengths are measured in the same unit and the relative speed is in a compatible distance-per-time unit.
Convert speed before applying the formula when train lengths are given in metres and time is required in seconds.
Quick Tricks
Do not add the speeds when one train overtakes another in the same direction. Subtract the smaller speed from the larger speed.
Complete overtaking occurs only after the rear end of the faster train passes the front end of the slower train. Hence, use L₁ + L₂, not the difference of lengths.
Multiplying a speed in km/h by 5/18 gives m/s, which can be directly used with lengths in metres.
Two Trains Overtaking Each Other Concepts
Relative Speed for Same-Direction Trains
If the faster train has speed S₁ and the slower train has speed S₂, then relative speed = S₁ − S₂, where S₁ > S₂. This relative motion treats the slower train as stationary and measures how quickly the faster train gains on it.
Distance Required for Complete Overtaking
At the beginning, the front of the faster train is behind the rear of the slower train. At the end, its rear must pass the front of the slower train. Thus, the relative distance is L₁ + L₂.
Finding Overtaking Time
Use T = (L₁ + L₂) ÷ (S₁ − S₂) for same-direction motion. If speeds are given in km/h and lengths in metres, convert the relative speed to m/s before calculating time in seconds.
Finding Speed or Train Length
From T = (L₁ + L₂) ÷ relative speed, relative speed = (L₁ + L₂) ÷ T and combined length = relative speed × T. If one length is known, subtract it from the combined length to find the other length.
Same Direction Versus Opposite Direction
For same-direction overtaking, relative speed = |S₁ − S₂|. When trains move in opposite directions and cross completely, relative speed = S₁ + S₂, and the crossing distance is still L₁ + L₂. Opposite-direction motion is usually called crossing rather than overtaking.
Two Trains Overtaking Each Other Data Table
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Relative-Speed Rules for Two Trains
The direction of motion determines whether train speeds are added or subtracted.
| Motion of trains | Relative speed | Distance for complete passage |
|---|---|---|
| Same direction; faster train overtakes slower train | |S₁ − S₂| | L₁ + L₂ |
| Opposite directions; trains cross each other | S₁ + S₂ | L₁ + L₂ |
Note: Keep distance and speed units compatible before calculating time.
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Two Trains Overtaking Each Other: Revision Points
Recall the direction rule, distance rule and unit conversion before solving a problem.
- For same-direction overtaking, relative speed is the difference of speeds.
- Overtaking distance is the sum of the two train lengths.
- Time = (sum of lengths) ÷ (difference of speeds).
- Convert km/h to m/s by multiplying by 5/18 when lengths are in metres.
- For opposite-direction crossing, add the speeds instead of subtracting them.
- If time and combined length are known, relative speed = combined length ÷ time.
Two Trains Overtaking Each Other FAQs
What is the formula for two trains overtaking each other?
For same-direction motion, time = (L₁ + L₂) ÷ |S₁ − S₂|. Here, L₁ and L₂ are train lengths, and S₁ and S₂ are their speeds in compatible units.
Why are the lengths of two trains added during overtaking?
The faster train must move forward by its own length plus the length of the slower train before it completely passes it. Therefore, overtaking distance = L₁ + L₂.
What is the relative speed when two trains move in the same direction?
Relative speed is the difference between their speeds. For example, at 90 km/h and 60 km/h, relative speed = 30 km/h.
Two trains are 120 m and 180 m long and move at 72 km/h and 54 km/h. How long do they take to overtake?
Relative speed = 18 km/h = 5 m/s. Combined length = 300 m, so time = 300 ÷ 5 = 60 seconds.
How do you convert a train speed from km/h to m/s?
Multiply the speed by 5/18. For example, 72 km/h = 72 × 5/18 = 20 m/s.
What happens if the two trains have equal speeds in the same direction?
Their relative speed is zero, so the faster train cannot overtake the other train. The overtaking time is therefore not finite.