Overtaking Problems Using Relative Speed with Examples
Overtaking problems using relative speed involve a faster vehicle or train passing a slower one moving in the same direction. The relative speed is the difference between their speeds, while the distance required for a complete overtake is the sum of their lengths. This page covers the formula, unit conversion, solving method and numerical examples.
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What Are Overtaking Problems Using Relative Speed?
For a complete overtake, the faster object must cover the sum of the lengths of both objects relative to the slower object. Therefore, for objects moving in the same direction: time = (length of faster object + length of slower object) ÷ (speed of faster object − speed of slower object). Speeds must be expressed in the same unit, usually metres per second when lengths are in metres.
Overtaking Problems Using Relative Speed Video Lessons
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Relative Speed in Same Direction
Learn how to calculate relative speed when two objects move in the same direction, including its application to overtaking problems.
Overtaking Problems Using Relative Speed Formula & Tricks
Important Formulas
Subtract the slower speed from the faster speed when both objects move in the same direction.
Here, L₁ and L₂ are the lengths of the two objects, and S₁ and S₂ are their speeds in the same units.
Use these conversions when lengths are given in metres and speeds are given in kilometres per hour.
This gives the actual distance travelled by the faster object during the overtaking interval.
Quick Tricks
For a complete overtake, add the lengths of both moving objects. Do not use only the length of the faster object.
When both objects move in the same direction, subtract their speeds before finding the time.
Convert km/h to m/s when the lengths are in metres. This avoids incorrect time units.
Overtaking Problems Using Relative Speed Concepts
Relative Speed for Same-Direction Motion
If the faster object has speed S₁ and the slower object has speed S₂, where S₁ > S₂, then relative speed = S₁ − S₂. The faster object gains this relative distance every unit of time.
Distance Required for a Complete Overtake
At the beginning, the front of the faster object is behind the rear of the slower object. At the end, its rear must be ahead of the slower object's front. Thus, the relative distance covered is L₁ + L₂.
Calculating Overtaking Time
Use time = (L₁ + L₂) ÷ (S₁ − S₂). Convert the speed difference into metres per second if the lengths are in metres. For example, two trains of lengths 120 m and 180 m move at 54 km/h and 36 km/h. Their relative speed is 18 km/h = 5 m/s, so time = 300 ÷ 5 = 60 seconds.
Vehicles or Objects with Negligible Length
For a car overtaking a stationary person treated as a point, distance = length of the car. If both objects have measurable lengths, add both lengths. The same rule applies to a train overtaking a pole, where the pole has negligible length.
Overtaking Problems Using Relative Speed Data Table
Use the structured data below while following the explanation and examples.
Overtaking Formula Cases
Relative speed and relative distance for common overtaking situations.
| Situation | Relative speed | Relative distance |
|---|---|---|
| Faster object overtakes slower object in same direction | S₁ − S₂ | L₁ + L₂ |
| Object passes a stationary point object | S | L |
| Faster object overtakes an object with negligible length | S₁ − S₂ | L₁ |
| Two objects move in opposite directions | S₁ + S₂ | L₁ + L₂ |
Note: The opposite-direction row describes crossing or passing, not ordinary overtaking. Use consistent units for speed, length and time.
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Overtaking Problems Using Relative Speed: Revision Points
Remember the direction rule, combined-length rule and unit conversion before calculating overtaking time.
- For same-direction motion, relative speed = faster speed − slower speed.
- For a complete overtake, relative distance = sum of the lengths of both objects.
- Overtaking time = relative distance ÷ relative speed.
- Convert km/h to m/s by multiplying by 5/18 when lengths are in metres.
- For a point object such as a pole, use only the length of the moving object.
- For opposite-direction passing, relative speed is the sum of the speeds, but this is a crossing case rather than same-direction overtaking.
Overtaking Problems Using Relative Speed FAQs
What is the overtaking problems using relative speed formula?
For objects moving in the same direction, time = (length of faster object + length of slower object) ÷ (speed of faster object − speed of slower object). Use compatible units.
Why are the lengths of both vehicles added in an overtaking problem?
A complete overtake requires the faster vehicle to move from behind the slower vehicle to fully ahead of it. Therefore, the relative distance is the sum of both vehicle lengths.
A 200 m train moves at 72 km/h and overtakes a 100 m train moving at 54 km/h. What is the overtaking time?
Relative speed = 72 − 54 = 18 km/h = 5 m/s. Time = (200 + 100) ÷ 5 = 60 seconds.
How do you convert a relative speed of 36 km/h into metres per second?
Multiply by 5/18: 36 × 5/18 = 10 m/s.
What happens if both objects have the same speed in the same direction?
Their relative speed is zero, so the faster object cannot overtake the slower object if their speeds remain equal.
How is overtaking a pole different from overtaking another train?
A pole has negligible length, so the relative distance is only the train's length. For another train, add both train lengths.