One-Line Definition
Distance is the total ground covered when moving at a given Speed for a given Time (Distance = Speed × Time). Relative speed measures closure speed between two moving bodies.
Highway Closure Mental Model
Two cars driving on a highway:
- • Driving towards each other (60 km/h & 40 km/h) → Close gap at 100 km/h (Add speeds!)
- • Driving in same direction → Faster car gains gap at 20 km/h (Subtract speeds!)
Time Speed Distance is heavily tested in CAT, SNAP, NMAT, and CET. Questions cover relative motion, train crossings, boats, circular tracks, and escalators:
Fast Unit Conversions (5/18 Rule)
Harmonic Mean Average Speed Formula:
SHORTCUT RULES & KEY FORMULAS
Relative Motion
Opposite Direction → Add speeds (S1 + S2)
Same Direction → Subtract speeds (|S1 - S2|)
Trains & Platforms
Crossing Pole → Distance = Train Length (L)
Crossing Platform → Distance = L_train + L_platform
River Current
Downstream D = Boat + Stream
Upstream U = Boat - Stream
Boat = (D + U)/2 ; Stream = (D - U)/2
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: A train 180m long crosses a telephone pole in 9 seconds. Find its speed in km/h.
Step 1: Speed in m/s = Distance / Time = 180 / 9 = 20 m/s.
Step 2: Convert to km/h = 20 × (18 / 5) = 72 km/h.
Q: Train A (120m, 54 km/h) and Train B (180m, 36 km/h) move in opposite directions. Time taken to cross each other?
Step 1: Convert speeds: 54 km/h = 15 m/s; 36 km/h = 10 m/s.
Step 2: Relative Speed = 15 + 10 = 25 m/s (Opposite directions add).
Step 3: Total Distance = 120 + 180 = 300m → Time = 300 / 25 = 12 seconds.
Q: A boat's downstream speed is 18 km/h and upstream speed is 10 km/h. Find the boat's speed in still water and stream speed.
Step 1: Boat Speed in still water = (D + U) / 2 = (18 + 10) / 2 = 14 km/h.
Step 2: Stream Speed = (D - U) / 2 = (18 - 10) / 2 = 4 km/h.
Mistake 1
Arithmetic Mean of Speeds
Taking (S1 + S2)/2 for average speed over equal distances instead of the harmonic mean 2·S1·S2 / (S1 + S2).
Mistake 2
Unit Mismatch
Mixing km/h with meters or seconds without converting using 5/18 or 18/5. Always align units first!
Mistake 3
Subtracting Opposite Speeds
Subtracting speeds when two objects move towards each other. Speeds ADD UP in opposite direction!
1. A car travels 150 km in 3 hours. Find its speed in m/s.
Answer: Speed = 150 / 3 = 50 km/h. In m/s = 50 × (5/18) = 13.89 m/s.
2. A man covers half his journey at 20 km/h and the remaining half at 30 km/h. Find his average speed for the whole journey.
Answer: Harmonic Mean = (2 × 20 × 30) / (20 + 30) = 1200 / 50 = 24 km/h.
3. Two trains 140m and 160m long run on parallel tracks at 60 km/h and 40 km/h respectively in the same direction. How long will the faster train take to pass the slower train?
Answer: Relative Speed = 60 - 40 = 20 km/h = 20 × (5/18) = 5.55 m/s. Total Distance = 140 + 160 = 300m → Time = 300 / (100/18) = 54 seconds.
FREQUENTLY ASKED QUESTIONS
❓ Why can't we take the normal average of speeds for a round trip?
Because time spent travelling at the slower speed is longer than at the faster speed. Average Speed = Total Distance / Total Time. When equal distances are covered, use the Harmonic Mean formula: Avg Speed = (2·S1·S2) / (S1 + S2).
❓ How do you convert km/h to m/s instantly?
To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.
❓ How do boats & streams formulas work?
Downstream speed (D) = Boat Speed + Stream Speed. Upstream speed (U) = Boat Speed - Stream Speed. Boat Speed = (D + U)/2, Stream Speed = (D - U)/2.