Introduction
Many motion problems involve two objects moving relative to each other - for example, two cars on a road, or a person and a train. The concept of relative speed tells you how fast the distance between them is changing, which makes meeting and overtaking problems straightforward.
Mastering relative speed helps you quickly determine meeting times, overtaking times, and closing/separating rates in both same-direction and opposite-direction scenarios.
Pattern: Relative Speed (Same or Opposite Direction)
Pattern
Key concept: When two objects move, treat one object as stationary by using the relative speed - the speed of one relative to the other.
- Same direction (both moving same way): Relative speed = |v₁ - v₂| (difference of speeds).
- Opposite direction (moving toward each other): Relative speed = v₁ + v₂ (sum of speeds).
Use consistent units (km/h with hours, m/s with seconds). For distances (like overtaking a length-L train), use: Time = Distance to be covered ÷ Relative speed.
Step-by-Step Example
Question
Two trains are 300 km apart and move towards each other at speeds 80 km/h and 70 km/h respectively. How long until they meet?
Solution
-
Step 1: Identify Given Values
Distance between trains = 300 km; speeds v₁ = 80 km/h, v₂ = 70 km/h. -
Step 2: Determine Relative Speed
They move in opposite directions (toward each other) → Relative speed = v₁ + v₂ = 80 + 70 = 150 km/h. -
Step 3: Compute Time to Meet
Time = Distance ÷ Relative speed = 300 ÷ 150 = 2 hours. -
Final Answer:
They will meet in 2 hours. -
Quick Check:
In 2 hours, first train covers 160 km and second covers 140 km → total 300 km ✅
Quick Variations
1. Overtaking a long object (train): Distance to cover = length of other object + initial gap (if any).
2. Meeting on circular tracks: treat direction (same/opposite) and use relative speed with laps/LCM for repeated meetings.
3. Converting units: if speeds in m/s and time given in seconds, use m and s consistently. Convert km/h ↔ m/s if needed.
4. When one object is stationary (v₂ = 0): relative speed = speed of moving object.
Trick to Always Use
- Step 1: Ask: "Same direction or opposite?" - this decides whether to add or subtract speeds.
- Step 2: Opposite → add speeds (v₁ + v₂). Same → subtract (|v₁ - v₂|).
- Step 3: Use Time = Distance ÷ Relative speed. Convert units first if necessary.
- Step 4: For overtaking trains/long objects, include their lengths in the distance to be covered.
Summary
Summary
Key takeaways:
- Relative speed makes two-body motion problems simple: add speeds for opposite-directions, subtract for same-direction.
- Always keep units consistent (km/h with hours, m/s with seconds).
- For overtaking, the distance to cover often equals the other object's length plus any initial gap.
- If unsure, compute each object's distance over the same time and equate or use total separation change method.
Hint: Opposite direction → add speeds (v1 + v2).
Common Mistakes: Subtracting speeds instead of adding when objects move toward each other.