Introduction
Meeting and overtaking problems are a core part of Time, Speed & Distance. They ask when two moving objects will meet (coming towards each other) or when one will catch up with and pass another (same direction). These problems become simple once you convert the scenario into a single formula using relative speed and the correct distance to be covered.
This pattern is important because it appears often in competitive exams and daily reasoning - trains, cars, runners - and is solved with the same small set of steps every time.
Pattern: Meeting and Overtaking Problems
Pattern
Key concept: Reduce the two-body problem to one-body by using relative speed and the specific distance that must be closed.
- Meeting (moving towards each other): Relative speed = v₁ + v₂. Time = Initial separation ÷ (v₁ + v₂).
- Overtaking (same direction): Relative speed = |v₁ - v₂| (faster - slower). Time = Distance to be covered ÷ (v_faster - v_slower).
- Distance to be covered: For meeting = initial gap; for overtaking an object (train/person) = length of object + any initial gap (if required).
- Unit consistency: Use km/h with hours or m/s with seconds - convert before computing.
Step-by-Step Example
Question
Two cars are 150 km apart and drive towards each other at 70 km/h and 50 km/h. When will they meet?
Solution
-
Step 1: Identify given values
Distance = 150 km; speeds v₁ = 70 km/h, v₂ = 50 km/h. -
Step 2: Decide case and compute relative speed
They move towards each other → Relative speed = v₁ + v₂ = 70 + 50 = 120 km/h. -
Step 3: Apply time formula
Time = Distance ÷ Relative speed = 150 ÷ 120 = 1.25 hours = 1 hour 15 minutes. -
Final Answer:
They meet in 1 hour 15 minutes. -
Quick Check:
In 1.25 h, first covers 87.5 km and second 62.5 km → total 150 km ✅
Quick Variations
1. Two objects meet after starting at different times: compute distances covered till meeting (use relative speed and adjust for head-start time).
2. Overtaking two long objects: Distance to cover = sum of both lengths (if one must fully clear the other).
3. Meeting on circular track: use relative speed; if looking for repeated meetings, consider LCM of lap-times or use time = circumference ÷ relative speed.
4. Units mixed (km/h and m/s): convert to common units before computing.
Trick to Always Use
- Step 1: Ask: "Are they moving towards each other (meeting) or same direction (overtaking)?"
- Step 2: For meeting → relative speed = v₁ + v₂. For overtaking → relative speed = v_faster - v_slower.
- Step 3: Figure the correct distance to be closed: initial gap, length(s), or head-start distance.
- Step 4: Ensure units match. Convert km/h ↔ m/s when time is required in seconds.
- Step 5: Time = Distance ÷ Relative speed. If needed, convert the answer into minutes/seconds for clarity.
Summary
Summary
- Identify whether the situation is meeting or overtaking to choose the correct relative speed formula.
- Always compute relative speed by adding (meeting) or subtracting (overtaking).
- Use the correct distance - initial separation, object length, or combined lengths - before applying the formula.
- Maintain consistent units (km/h ↔ m/s) and convert time to minutes or seconds when required.
Example to remember:
“Meeting → add speeds, Overtaking → subtract speeds, then Time = Distance ÷ Relative Speed.”
Hint: Meeting → Add speeds to get relative speed.
Common Mistakes: Subtracting speeds instead of adding for opposite directions.