Time, Speed and Distance

Train problems boil down to: time = (total length) / (relative speed).

Scenario 1: Train crossing a stationary point/pole.
Total length = train length L. Relative speed = train speed v.
t = L / v

Scenario 2: Train crossing a stationary platform/tunnel.
Total length = train length + platform length = L + P. Speed = v.
t = (L + P) / v

Scenario 3: Two trains in opposite directions.
Relative speed = sum of speeds = v₁ + v₂. Total length = sum of lengths.
t = (L₁ + L₂) / (v₁ + v₂)

Scenario 4: Two trains in same direction (faster overtakes slower).
Relative speed = difference of speeds = v₁ − v₂. Total length = sum of lengths.
t = (L₁ + L₂) / (v₁ − v₂)

Unit conversion: 1 km/h = 5/18 m/s. 1 m/s = 18/5 km/h.

Worked example. A 200 m train moving at 72 km/h overtakes a 300 m train moving at 54 km/h in the same direction. Time taken?

Convert speeds: 72 km/h = 20 m/s. 54 km/h = 15 m/s. Relative speed = 5 m/s. Total length = 500 m. Time = 500/5 = 100 seconds.

CORE RELATION:

Distance = Speed × Time.
Speed = Distance / Time.
Time = Distance / Speed.

UNIT CONVERSIONS:

• 1 km/h = 5/18 m/s. (Divide by 18, multiply by 5.)
• 1 m/s = 18/5 km/h.

Memorize: 36 km/h = 10 m/s.

AVERAGE SPEED

NOT (s₁ + s₂)/2 always.

For equal DISTANCES: avg speed = 2s₁s₂/(s₁+s₂) (harmonic mean).
For equal TIMES: avg speed = (s₁+s₂)/2 (arithmetic mean).

Example: a car goes 60 km at 30 km/h and returns at 60 km/h. Avg speed?
= 2(30)(60)/(30+60) = 3600/90 = 40 km/h. (NOT 45.)

RELATIVE SPEED

Same direction: difference of speeds.
Opposite direction: sum of speeds.

Used for: trains crossing, two runners catching up, etc.

TRAINS:

Train crossing a stationary man/pole:
Time = length of train / speed of train.

Train crossing a platform/bridge:
Time = (length of train + length of bridge) / speed of train.

Two trains crossing each other:

• Same direction: time = (L₁+L₂)/|v₁−v₂|.
• Opposite direction: time = (L₁+L₂)/(v₁+v₂).

BOATS & STREAMS (covered in Pack 12; quick recap):
Downstream = b + s; Upstream = b − s.

CIRCULAR / RACE TRACKS

Two runners on circular track:

• Same direction: time between meetings = L / (v₁ − v₂) (if v₁ > v₂).
• Opposite direction: time = L / (v₁ + v₂).

EXAMPLES

Q1. A car travels 240 km at 60 km/h, then returns at 40 km/h. Find avg speed.
= 2(60)(40)/(60+40) = 4800/100 = 48 km/h.

Q2. A train 150 m long crosses a pole in 15 s. Speed?
v = 150/15 = 10 m/s = 36 km/h.

Q3. A train 200 m long crosses a 100 m platform in 30 s. Speed?
v = (200+100)/30 = 10 m/s = 36 km/h.

Q4. Two trains in opposite directions, lengths 100 m and 150 m, speeds 36 and 54 km/h. Time to cross?
v_rel = 36 + 54 = 90 km/h = 25 m/s.
Distance = 100 + 150 = 250 m. Time = 250/25 = 10 s.

Q5. A starts walking at 4 km/h. After 30 min, B starts cycling at 8 km/h on same route, same direction. When does B catch A?
Head start of A: 4 × 0.5 = 2 km.
Relative speed = 8 − 4 = 4 km/h.
Time to catch up = 2/4 = 0.5 hr = 30 min after B starts.

Q6. Train passes a man walking 6 km/h in same direction in 30 s; passes him in opposite direction in 10 s. Find train speed and length.
Let train speed = v, length = L (km).
Same direction: L = (v−6) × 30/3600 = (v−6)/120.
Opposite: L = (v+6) × 10/3600 = (v+6)/360.
Equate: 3(v−6) = v+6 → 3v − 18 = v + 6 → v = 12 km/h.
L = (12+6)/360 = 18/360 = 1/20 km = 50 m.

TIPS:

• Convert all units to same system (m/s or km/h) before computing.
• For "average speed", check whether it's equal distance or equal time.
• "Crossing pole" = train length only; "crossing platform/bridge" = sum.
• Relative speed: train catching another in same direction is much slower than meeting in opposite directions.

CORE RELATION:

Distance = Speed × Time.
Speed = Distance / Time.
Time = Distance / Speed.

UNIT CONVERSIONS:

• 1 km/h = 5/18 m/s. (Divide by 18, multiply by 5.)
• 1 m/s = 18/5 km/h.

Memorize: 36 km/h = 10 m/s.

AVERAGE SPEED

NOT (s₁ + s₂)/2 always.

For equal DISTANCES: avg speed = 2s₁s₂/(s₁+s₂) (harmonic mean).
For equal TIMES: avg speed = (s₁+s₂)/2 (arithmetic mean).

Example: a car goes 60 km at 30 km/h and returns at 60 km/h. Avg speed?
= 2(30)(60)/(30+60) = 3600/90 = 40 km/h. (NOT 45.)

RELATIVE SPEED

Same direction: difference of speeds.
Opposite direction: sum of speeds.

Used for: trains crossing, two runners catching up, etc.

TRAINS:

Train crossing a stationary man/pole:
Time = length of train / speed of train.

Train crossing a platform/bridge:
Time = (length of train + length of bridge) / speed of train.

Two trains crossing each other:

• Same direction: time = (L₁+L₂)/|v₁−v₂|.
• Opposite direction: time = (L₁+L₂)/(v₁+v₂).

BOATS & STREAMS (covered in Pack 12; quick recap):
Downstream = b + s; Upstream = b − s.

CIRCULAR / RACE TRACKS

Two runners on circular track:

• Same direction: time between meetings = L / (v₁ − v₂) (if v₁ > v₂).
• Opposite direction: time = L / (v₁ + v₂).

EXAMPLES

Q1. A car travels 240 km at 60 km/h, then returns at 40 km/h. Find avg speed.
= 2(60)(40)/(60+40) = 4800/100 = 48 km/h.

Q2. A train 150 m long crosses a pole in 15 s. Speed?
v = 150/15 = 10 m/s = 36 km/h.

Q3. A train 200 m long crosses a 100 m platform in 30 s. Speed?
v = (200+100)/30 = 10 m/s = 36 km/h.

Q4. Two trains in opposite directions, lengths 100 m and 150 m, speeds 36 and 54 km/h. Time to cross?
v_rel = 36 + 54 = 90 km/h = 25 m/s.
Distance = 100 + 150 = 250 m. Time = 250/25 = 10 s.

Q5. A starts walking at 4 km/h. After 30 min, B starts cycling at 8 km/h on same route, same direction. When does B catch A?
Head start of A: 4 × 0.5 = 2 km.
Relative speed = 8 − 4 = 4 km/h.
Time to catch up = 2/4 = 0.5 hr = 30 min after B starts.

Q6. Train passes a man walking 6 km/h in same direction in 30 s; passes him in opposite direction in 10 s. Find train speed and length.
Let train speed = v, length = L (km).
Same direction: L = (v−6) × 30/3600 = (v−6)/120.
Opposite: L = (v+6) × 10/3600 = (v+6)/360.
Equate: 3(v−6) = v+6 → 3v − 18 = v + 6 → v = 12 km/h.
L = (12+6)/360 = 18/360 = 1/20 km = 50 m.

TIPS:

• Convert all units to same system (m/s or km/h) before computing.
• For "average speed", check whether it's equal distance or equal time.
• "Crossing pole" = train length only; "crossing platform/bridge" = sum.
• Relative speed: train catching another in same direction is much slower than meeting in opposite directions.