Time, Speed & Distance – Complete Short Notes for Competitive Exams

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1. Basic Relationship

The golden formula that connects all three quantities:

$$\boxed{\text{Speed} = \frac{\text{Distance}}{\text{Time}}}$$ $$\boxed{\text{Time} = \frac{\text{Distance}}{\text{Speed}}}$$ $$\boxed{\text{Distance} = \text{Speed} × \text{Time}}$$

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2. Standard Units

Quantity	SI Unit	Conversion
Distance	Kilometer (km) or Meter (m)	1 km = 1000 m
Time	Hour or Second	1 hour = 3600 sec
Speed	km/hr or m/sec	1 m/s = 3.6 km/hr

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Conversion Formulas

• To convert km/hr → m/s, multiply by 5/18

• To convert m/s → km/hr, multiply by 18/5

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Example:

72 km/hr = 72 × (5/18) = 20 m/s
10 m/s = 10 × (18/5) = 36 km/hr

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3. Average Speed

When distance is same but speed varies:

$$\text{Average Speed} = \frac{2S₁S₂}{S₁ + S₂}$$

When distances differ:

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

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Example:

A person travels 60 km at 30 km/hr and returns at 90 km/hr.

$$\text{Avg Speed} = \frac{2×30×90}{30 + 90} = 45 km/hr$$

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4. Relative Speed

Used when two objects move towards or away from each other.

Direction	Formula
Opposite	Add the speeds
Same	Subtract the speeds

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Examples:

• Two trains of 40 km/hr and 20 km/hr moving towards each other → Relative Speed = 40 + 20 = 60 km/hr

• If in same direction → Relative Speed = 40 − 20 = 20 km/hr

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5. Important Concept: Time Taken to Meet

If two objects start from two points and move towards each other:

$$\text{Time} = \frac{\text{Distance between them}}{\text{Sum of their speeds}}$$

If moving in same direction:

$$\text{Time} = \frac{\text{Distance between them}}{\text{Difference of their speeds}}$$

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6. Concept of Speed Ratio

When distances are equal:

$$\text{Speed Ratio} = \frac{1}{\text{Time Ratio}}$$

When times are equal:

$$\text{Distance Ratio} = \text{Speed Ratio}$$

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Example:

If a car covers same distance in 2 hours and a bike in 3 hours,
Speed ratio = 3 : 2

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7. Shortcut Relation (Equal Distance Case)

If same distance covered at two speeds S₁, S₂, then:

Time ratio = S₂ : S₁
Average speed = 2S₁S₂ / (S₁ + S₂)

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8. Problems on Trains

When a train passes:

• A pole:
  Distance = Length of train
  Time = Time to pass pole
  Speed = Distance / Time

• A platform:
  Distance = Length of train + Length of platform
  Time = Time to cross platform
  Speed = Distance / Time

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Example:

A train 200 m long crosses a platform 300 m in 20 sec.
Speed = (200 + 300)/20 = 25 m/s = 25 × 3.6 = 90 km/hr

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9. Relative Speed (Trains on Tracks)

• Trains moving in opposite direction:
  Relative Speed = (S₁ + S₂)

• Trains moving in same direction:
  Relative Speed = (S₁ − S₂)

Then:

$$\text{Time to cross each other} = \frac{\text{Sum of lengths}}{\text{Relative Speed}}$$

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10. Boat and Stream Problems

Type	Formula
Speed in still water (B)	—
Speed of stream (S)	—
Downstream speed	(B + S)
Upstream speed	(B − S)

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Key Formulas

$$\text{Speed in Still Water} = \frac{\text{Downstream + Upstream}}{2}$$ $$\text{Speed of Stream} = \frac{\text{Downstream − Upstream}}{2}$$

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Example:

Downstream speed = 12 km/hr, Upstream speed = 8 km/hr
→ B = (12 + 8)/2 = 10 km/hr
→ S = (12 − 8)/2 = 2 km/hr

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11. Important Relation – Time Difference

If the same distance is covered at two speeds S₁, S₂:

$$\text{Difference in Time}=D(\frac{1}{S_1}-\frac{1}{S_2})$$

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Example:

Distance = 120 km, Speeds = 40 & 60 km/hr
Time diff = 120(1/40 − 1/60) = 120(1/120) = 1 hr

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12. Motion in Circular Track

If two runners start from same point and move in same direction:

$$\text{Time to meet}=\frac{\text{Length of Track}}{\text{Difference of Speeds}}$$

If they move in opposite directions:

$$\text{Time to meet}=\frac{\text{Length of Track}}{\text{Sum of Speeds}}$$

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13. Points of Meeting in Circular Track

If two persons run around a circular track of length L:

• Same direction:
  Number of meetings = (Relative speed × Total time) / L

• Opposite direction:
  Meetings are more frequent as speed adds up.

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14. Unit Conversion Shortcuts

From	To
km/hr → m/s	× 5/18
m/s → km/hr	× 18/5
km/min → m/s	× (1000/60) × (1/1) = 16.67
km/hr → m/min	× (1000/60) = 16.67

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15. Key Practice Types

Type	Formula
Speed, Time, Distance	S = D/T
Average Speed (equal D)	2S₁S₂/(S₁ + S₂)
Relative Speed	S₁ ± S₂
Boat/Stream	B ± S
Train crosses pole	L / T
Train crosses platform	(L₁ + L₂) / T
Time difference	D(1/S₁ − 1/S₂)

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16. Common Shortcut Tricks

✅ If distance same → Time ∝ 1/Speed
✅ Ratio of speeds = Inverse of ratio of times
✅ Always convert speeds to same units before using formulas
✅ For circular/loop tracks → relative speed concept applies
✅ If object returns to same point → use average speed formula

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17. Practice Examples

Q1. A car travels 150 km in 3 hrs. Find speed.
→ S = D/T = 150/3 = 50 km/hr

Q2. A train 120 m long crosses a man in 8 sec.
→ S = 120/8 = 15 m/s = 15×3.6 = 54 km/hr

Q3. A boat goes 24 km downstream in 3 hrs and upstream in 4 hrs.
→ D.S. = 8 km/hr, U.S. = 6 km/hr
→ B = (8 + 6)/2 = 7 km/hr, S = (8 − 6)/2 = 1 km/hr

Q4. A man walks at 6 km/hr and runs at 9 km/hr for equal distance.
→ Avg speed = 2×6×9 / (6 + 9) = 108/15 = 7.2 km/hr

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18. Quick Revision Summary

Concept	Key Formula
Basic	S = D/T
Unit Conversion	1 m/s = 3.6 km/hr
Average Speed (equal D)	2S₁S₂/(S₁ + S₂)
Relative Speed	S₁ ± S₂
Boat Stream	B ± S
Train vs Platform	(L₁ + L₂)/T
Time Diff	D(1/S₁ − 1/S₂)

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19. Real-life Concept Links

• Vehicles (cars, bikes, trains) → average speed

• Rivers → boat & stream

• Athletics → circular track, relative speed

• Time management → distance = progress, speed = efficiency

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✅ In One Line:

“Speed tells how fast, Time tells how long, and Distance tells how far — all tied by S = D/T.”