DEFINITION
One-Line Definition
Relative Speed is the rate at which the distance between two moving bodies decreases (approaching) or increases (separating), measured relative to one another.
Meeting Time = Initial Distance Gap / Relative Speed
CORE INTUITION ⚡
Two-Body Motion Model
To simplify two moving objects, freeze one object and adjust the speed of the other:
- • Moving OPPOSITE (towards each other) → Speeds ADD: S1 + S2
- • Moving SAME direction (chasing/overtaking) → Speeds SUBTRACT: |S1 - S2|
Always equalize the starting times before applying relative speed formulas!
💡 WHY THIS CONCEPT MATTERS & REAL-LIFE APPLICATIONS
Relative Speed is the foundation behind train crossings, police-thief chases, circular tracks, and multi-body motion in CAT & MBA CET. Click on any connected topic to jump directly to its Vault page:
Where Is Relative Speed Used?
🚔 Police-Thief Pursuit
✈️ Air Traffic Flight Paths
🚘 Highway Overtaking
🏎️ Circular Track Racing
📐 RELATIVE SPEED FORMULAS & CASES
THE GENERAL RELATIVE SPEED FORMULA
Time to Meet / Catch Up = Initial Gap / Relative Speed
t = Distance Gap / S_rel
Relative Speed Rules by Direction
OPPOSITE DIRECTION (HEAD-ON)
S_rel = S1 + S2
Objects approach faster → ADD speeds
SAME DIRECTION (CHASE / OVERTAKE)
S_rel = |S1 - S2|
Faster object closes gap → SUBTRACT speeds
📝 SOLVED EXAMPLES (LEVEL 0 TO ADVANCED)
EASY • EXAMPLE 1
Two cars A and B start from two points 200 km apart moving towards each other at 60 km/h and 40 km/h. Find when they meet.
Solution:
Opposite Direction => Relative Speed = S1 + S2 = 60 + 40 = 100 km/h
Initial Distance Gap = 200 km
Meeting Time = Distance Gap / S_rel = 200 / 100 = 2 hours
Answer: 2 hours
MEDIUM • EXAMPLE 2 (POLICE-THIEF CHASE)
A thief steals a car at 1:00 PM and drives at 60 km/h. The theft is discovered at 2:00 PM and police start chasing at 80 km/h. When will police catch the thief?
Solution:
Head Start Time = 1:00 PM to 2:00 PM = 1 hour
Thief's Distance in 1 hr = 60 × 1 = 60 km (Head Start Gap)
From 2:00 PM onwards, both move in SAME direction:
Relative Speed = 80 - 60 = 20 km/h
Time to Catch = Head Start Gap / S_rel = 60 / 20 = 3 hours
Catch Time = 2:00 PM + 3 hours = 5:00 PM
Answer: 5:00 PM
HARD • EXAMPLE 3 (DIFFERENT START TIMES)
A starts from Town P to Town Q at 8:00 AM at 40 km/h. B starts from Town Q to Town P at 9:00 AM at 60 km/h. Total distance PQ = 340 km. Find when they meet.
Solution:
From 8:00 AM to 9:00 AM (1 hr), A travels = 40 × 1 = 40 km
Remaining Distance at 9:00 AM = 340 - 40 = 300 km
From 9:00 AM onwards, both move in OPPOSITE directions:
Relative Speed = 40 + 60 = 100 km/h
Time to meet from 9:00 AM = 300 / 100 = 3 hours
Meeting Time = 9:00 AM + 3 hours = 12:00 PM (Noon)
Answer: 12:00 PM (Noon)
⚠️ COMMON MISTAKES TO AVOID
❌ Mistake 1: Subtracting Speeds in Opposite Directions
When two objects move towards each other, the distance between them reduces faster! Always ADD speeds (S1 + S2) for opposite directions.
❌ Mistake 2: Forgetting Head-Start Distance
If Person A starts at 8 AM and Person B starts at 9 AM, do NOT divide the total distance by relative speed at 8 AM! Calculate how far A travels by 9 AM first.
❌ Mistake 3: Misidentifying Same vs Opposite Directions
Overtaking, chasing, or running behind someone = Same Direction (|S1 - S2|). Approaching head-on = Opposite Direction (S1 + S2).
🚀 CAT & MBA CET SHORTCUTS
⚡ Shortcut 1: Meeting Point Distance Ratio
When two bodies move towards each other from two ends at the same time:
• The distance covered by A and B until they meet is directly proportional to their speeds: D1 : D2 = S1 : S2.
• The distance covered by A and B until they meet is directly proportional to their speeds: D1 : D2 = S1 : S2.
⚡ Shortcut 2: Equalize Starting Times First
Always bring both objects to the exact same starting timestamp before applying t = Gap / Srel. This prevents 95% of time-of-meeting errors!
🎯 PRACTICE QUESTIONS
QUESTION 1 • BASIC
Two trains start from stations 300 km apart moving towards each other at 50 km/h and 70 km/h. Find when they meet.
Relative Speed = 50 + 70 = 120 km/h.
Meeting Time = 300 / 120 = 2.5 hours (2 hrs 30 mins).
Meeting Time = 300 / 120 = 2.5 hours (2 hrs 30 mins).
QUESTION 2 • BASIC
A police jeep at 90 km/h chases a thief's car traveling at 70 km/h who is 100 km ahead. Find the time taken to catch the thief.
Relative Speed (same direction) = 90 - 70 = 20 km/h.
Time to catch = 100 / 20 = 5 hours.
Time to catch = 100 / 20 = 5 hours.
QUESTION 3 • MODERATE
Person A starts walking at 7:00 AM at 30 km/h. Person B starts from the same point at 9:00 AM at 50 km/h in the same direction. At what time will B catch A?
Head start distance of A (7 AM to 9 AM) = 30 × 2 = 60 km.
Relative Speed = 50 - 30 = 20 km/h.
Time to catch from 9 AM = 60 / 20 = 3 hours.
Catch time = 9:00 AM + 3 hours = 12:00 PM (Noon).
Relative Speed = 50 - 30 = 20 km/h.
Time to catch from 9 AM = 60 / 20 = 3 hours.
Catch time = 9:00 AM + 3 hours = 12:00 PM (Noon).
QUESTION 4 • ADVANCED
Two towns A and B are 450 km apart. Train 1 leaves A at 8:00 AM at 60 km/h towards B. Train 2 leaves B at 9:00 AM at 70 km/h towards A. At what exact time do they meet?
From 8 AM to 9 AM (1 hr), Train 1 covers 60 km.
Remaining distance at 9 AM = 450 - 60 = 390 km.
Relative Speed = 60 + 70 = 130 km/h.
Time from 9 AM = 390 / 130 = 3 hours.
Meeting time = 9:00 AM + 3 hours = 12:00 PM (Noon).
Remaining distance at 9 AM = 450 - 60 = 390 km.
Relative Speed = 60 + 70 = 130 km/h.
Time from 9 AM = 390 / 130 = 3 hours.
Meeting time = 9:00 AM + 3 hours = 12:00 PM (Noon).
❓ FREQUENTLY ASKED QUESTIONS
Q: When do we add speeds and when do we subtract speeds in Relative Speed?
Add speeds (S1 + S2) when objects move in OPPOSITE directions (approaching each other). Subtract speeds |S1 - S2| when objects move in the SAME direction (overtaking or chasing).
Q: How do we handle relative speed problems where objects start at different times?
First calculate the distance covered by the early starter during the head-start time. Subtract this distance from the total gap, then divide the remaining distance by the relative speed from the moment both objects start moving.
Q: Does Relative Speed apply to circular tracks?
Yes! On a circular track of length L, the time taken for two runners to meet for the first time is L / (S1 + S2) when running in opposite directions, and L / |S1 - S2| when running in the same direction.