📚 Basic Concepts

Key Definitions

• Speed: Distance covered per unit time
• Time: Duration taken to cover a distance
• Distance: Length of path covered
• Velocity: Speed with direction (for competitive exams, often treated same as speed)

Fundamental Formulas

Speed = Distance ÷ Time
Distance = Speed × Time  
Time = Distance ÷ Speed

Units and Conversions

• Speed Units: km/hr, m/s, mph
• Key Conversion: 1 km/hr = 5/18 m/s
• To convert:
  • km/hr to m/s: multiply by 5/18
  • m/s to km/hr: multiply by 18/5

Important Principles

1. Average Speed = Total Distance ÷ Total Time
2. Relative Speed (same direction) = |S₁ – S₂|
3. Relative Speed (opposite direction) = S₁ + S₂

🔄 Essential Formulas

Basic Speed Conversions

• km/hr to m/s: Speed × (5/18)
• m/s to km/hr: Speed × (18/5)
• Quick trick: km/hr to m/s → divide by 3.6

Relative Speed

• Same Direction: Faster speed – Slower speed
• Opposite Direction: Speed₁ + Speed₂
• Meeting Time = Distance between them ÷ Relative speed

Average Speed

• For two speeds S₁ and S₂ over equal distances: Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

📖 Type 1: Basic Speed, Time & Distance

Example 1

Problem: A car travels 240 km in 4 hours. What is its speed in km/hr and m/s?

Solution:

• Speed = Distance ÷ Time = 240 ÷ 4 = 60 km/hr
• In m/s = 60 × (5/18) = 60 × 5/18 = 16.67 m/s

Example 2

Problem: A train runs at 54 km/hr. How much distance will it cover in 20 minutes?

Solution:

• Time = 20 minutes = 20/60 = 1/3 hour
• Distance = Speed × Time = 54 × (1/3) = 18 km

Example 3

Problem: A person walks 15 km in 3 hours. At what speed should he walk to cover 25 km in 4 hours?

Solution:

• Current speed = 15 ÷ 3 = 5 km/hr
• Required speed = 25 ÷ 4 = 6.25 km/hr
• Increase needed = 6.25 – 5 = 1.25 km/hr

📖 Type 2: Relative Speed

Example 4

Problem: Two trains are moving in opposite directions at 60 km/hr and 40 km/hr. They cross each other in 9 seconds. Find the sum of their lengths.

Solution:

• Relative speed = 60 + 40 = 100 km/hr
• In m/s = 100 × (5/18) = 250/9 m/s
• Combined length = Speed × Time = (250/9) × 9 = 250 meters

Example 5

Problem: A faster train overtakes a slower train. Faster train: 80 km/hr, slower: 50 km/hr. Time to overtake = 36 seconds. Find combined length.

Solution:

• Relative speed = 80 – 50 = 30 km/hr
• In m/s = 30 × (5/18) = 25/3 m/s
• Combined length = (25/3) × 36 = 300 meters

Example 6

Problem: Two cars start from same point in opposite directions at 30 km/hr and 40 km/hr. After how much time will they be 210 km apart?

Solution:

• Relative speed = 30 + 40 = 70 km/hr
• Time = Distance ÷ Speed = 210 ÷ 70 = 3 hours

📖 Type 3: Trains Problems

Example 7

Problem: A 150m long train crosses a 250m platform in 20 seconds. Find the speed of train.

Solution:

• Total distance = Length of train + Length of platform = 150 + 250 = 400m
• Speed = 400 ÷ 20 = 20 m/s
• In km/hr = 20 × (18/5) = 72 km/hr

Example 8

Problem: A train 120m long running at 36 km/hr crosses a man running at 9 km/hr in opposite direction. Time to cross?

Solution:

• Train speed = 36 km/hr, Man speed = 9 km/hr
• Relative speed = 36 + 9 = 45 km/hr = 45 × (5/18) = 12.5 m/s
• Time = Distance ÷ Speed = 120 ÷ 12.5 = 9.6 seconds

Example 9

Problem: Two trains 140m and 160m long are running at 42 km/hr and 30 km/hr in same direction. Time for faster to overtake slower?

Solution:

• Relative speed = 42 – 30 = 12 km/hr = 12 × (5/18) = 10/3 m/s
• Combined length = 140 + 160 = 300m
• Time = 300 ÷ (10/3) = 300 × 3/10 = 90 seconds

📖 Type 4: Boats and Streams

Example 10

Problem: Speed of boat in still water = 15 km/hr, stream speed = 3 km/hr. Find downstream and upstream speeds.

Solution:

• Downstream speed = Boat speed + Stream speed = 15 + 3 = 18 km/hr
• Upstream speed = Boat speed – Stream speed = 15 – 3 = 12 km/hr

Example 11

Problem: A boat takes 2 hours downstream and 3 hours upstream to cover 24 km each way. Find boat speed and stream speed.

Solution:

• Let boat speed = b, stream speed = s
• Downstream: 24 = (b + s) × 2 → b + s = 12
• Upstream: 24 = (b – s) × 3 → b – s = 8
• Adding: 2b = 20 → b = 10 km/hr
• Subtracting: 2s = 4 → s = 2 km/hr

Example 12

Problem: Downstream speed = 20 km/hr, upstream speed = 10 km/hr. Find still water speed and stream speed.

Solution:

• Still water speed = (Downstream + Upstream) ÷ 2 = (20 + 10) ÷ 2 = 15 km/hr
• Stream speed = (Downstream – Upstream) ÷ 2 = (20 – 10) ÷ 2 = 5 km/hr

📖 Type 5: Average Speed

Example 13

Problem: A car covers first 100 km at 50 km/hr and next 100 km at 40 km/hr. Find average speed.

Solution:

• Time for first 100 km = 100 ÷ 50 = 2 hours
• Time for next 100 km = 100 ÷ 40 = 2.5 hours
• Total distance = 200 km, Total time = 4.5 hours
• Average speed = 200 ÷ 4.5 = 44.44 km/hr

Example 14

Problem: A person travels equal distances at 30 km/hr, 40 km/hr, and 60 km/hr. Find average speed.

Solution:

• Let each distance = d
• Time₁ = d/30, Time₂ = d/40, Time₃ = d/60
• Total distance = 3d
• Total time = d/30 + d/40 + d/60 = d(4+3+2)/120 = 9d/120 = 3d/40
• Average speed = 3d ÷ (3d/40) = 40 km/hr

Example 15

Problem: Using formula for two equal distances at speeds S₁ and S₂: Average = (2×S₁×S₂)/(S₁+S₂)

For S₁ = 60 km/hr, S₂ = 40 km/hr: Average = (2×60×40)/(60+40) = 4800/100 = 48 km/hr

📖 Type 6: Races and Games

Example 16

Problem: In a 1000m race, A beats B by 100m. If A’s speed is 10 m/s, find B’s speed.

Solution:

• When A covers 1000m, B covers 900m
• Time taken by A = 1000 ÷ 10 = 100 seconds
• B’s speed = 900 ÷ 100 = 9 m/s

Example 17

Problem: A gives B a start of 20m in a 100m race and still beats him by 5 seconds. If A’s speed is 8 m/s, find B’s speed.

Solution:

• A covers 100m in 100 ÷ 8 = 12.5 seconds
• B covers 80m in 12.5 + 5 = 17.5 seconds
• B’s speed = 80 ÷ 17.5 = 4.57 m/s

Example 18

Problem: In a circular track of 400m, A and B start from same point in same direction. A’s speed = 8 m/s, B’s speed = 6 m/s. When will A lap B?

Solution:

• Relative speed = 8 – 6 = 2 m/s
• Time to gain one full lap = 400 ÷ 2 = 200 seconds

📖 Type 7: Meeting Point Problems

Example 19

Problem: Two persons A and B are 120 km apart. They start towards each other at 30 km/hr and 20 km/hr. Where and when do they meet?

Solution:

• Combined speed = 30 + 20 = 50 km/hr
• Time to meet = 120 ÷ 50 = 2.4 hours
• Distance covered by A = 30 × 2.4 = 72 km from A’s starting point
• Distance covered by B = 20 × 2.4 = 48 km from B’s starting point

Example 20

Problem: A starts from P to Q at 60 km/hr. After 2 hours, B starts from Q to P at 80 km/hr. Distance PQ = 480 km. When do they meet?

Solution:

• In 2 hours, A covers = 60 × 2 = 120 km
• Remaining distance = 480 – 120 = 360 km
• Combined speed = 60 + 80 = 140 km/hr
• Time to meet after B starts = 360 ÷ 140 = 18/7 hours
• Total time from A’s start = 2 + 18/7 = 32/7 hours ≈ 4.57 hours

📖 Type 8: Circular Motion

Example 21

Problem: Two runners on a 400m circular track start together. A runs at 8 m/s, B at 6 m/s in same direction. After how much time will A be exactly one lap ahead?

Solution:

• Relative speed = 8 – 6 = 2 m/s
• Time for A to gain 400m = 400 ÷ 2 = 200 seconds

Example 22

Problem: On a 200m circular track, two runners start from opposite points. A at 5 m/s, B at 3 m/s in same direction. When do they first meet?

Solution:

• Initial separation = 100m (half track)
• Relative speed = 5 – 3 = 2 m/s
• Time to meet = 100 ÷ 2 = 50 seconds

🎯 Quick Tips for Competitive Exams

Speed Conversion Shortcuts

• km/hr to m/s: Multiply by 5/18 or divide by 3.6
• m/s to km/hr: Multiply by 18/5 or multiply by 3.6
• Common conversions to remember:
  • 18 km/hr = 5 m/s
  • 36 km/hr = 10 m/s
  • 54 km/hr = 15 m/s
  • 72 km/hr = 20 m/s

Train Problems Quick Rules

1. Crossing a pole/man: Distance = Length of train
2. Crossing a platform: Distance = Length of train + Length of platform
3. Two trains crossing: Distance = Sum of both train lengths

Boats and Streams Formulas

• Still water speed = (Downstream + Upstream) ÷ 2
• Stream speed = (Downstream – Upstream) ÷ 2
• Time ratio (Down:Up) = (b-s) : (b+s) where b=boat speed, s=stream speed

Average Speed Shortcuts

• For equal distances: Use harmonic mean formula
• For equal times: Simple arithmetic mean
• Remember: Average speed ≠ Average of speeds

Relative Speed Memory Tricks

• Same direction: Subtract (like chasing)
• Opposite direction: Add (like collision)
• Meeting problems: Always use combined speed

📊 Important Patterns & Ratios

Common Speed Ratios

• If speeds are in ratio 3:4, times are in ratio 4:3
• If A is 25% faster than B, speed ratio = 5:4, time ratio = 4:5
• If A is 20% slower than B, speed ratio = 4:5, time ratio = 5:4

Distance-Time Relationships

• Distance ∝ Speed (when time is constant)
• Distance ∝ Time (when speed is constant)
• Speed ∝ 1/Time (when distance is constant)

Train Length Patterns

• Most train problems use lengths: 100m, 120m, 150m, 200m, 250m
• Platform lengths: 150m, 200m, 250m, 300m, 400m

📝 Practice Problems

Set A – Basic Problems

1. Convert 108 km/hr to m/s
2. A car travels 180 km in 3 hours. Find speed in m/s
3. At 45 km/hr, how much distance in 24 minutes?

Set B – Trains & Relative Speed

4. 200m train crosses 150m platform in 14 seconds. Find speed in km/hr
5. Two trains 120m and 80m long, speeds 54 km/hr and 36 km/hr, same direction. Overtaking time?
6. Train 150m long crosses a man in 18 seconds. Speed of train?

Set C – Boats & Streams

7. Boat speed 12 km/hr, stream 2 km/hr. Time to go 28 km downstream?
8. Downstream 15 km/hr, upstream 9 km/hr. Still water and stream speeds?
9. 24 km downstream in 2 hrs, same distance upstream in 3 hrs. Find speeds.

Set D – Average Speed & Complex

10. First half distance at 40 km/hr, second half at 60 km/hr. Average speed?
11. A and B are 300 km apart, approach each other at 80 km/hr and 70 km/hr. Meeting time and point?
12. Circular track 500m, A at 10 m/s, B at 8 m/s, same direction from same point. When does A lap B?

🏆 Answer Key

Set A: 1) 30 m/s 2) 16.67 m/s 3) 18 km Set B: 4) 90 km/hr 5) 40 seconds 6) 30 km/hr Set C: 7) 2 hours 8) Still: 12 km/hr, Stream: 3 km/hr 9) Boat: 10 km/hr, Stream: 2 km/hr Set D: 10) 48 km/hr 11) 2 hours, 160 km from A’s start 12) 250 seconds

🔍 Common Exam Mistakes to Avoid

1. Unit confusion: Always check if answer needed in km/hr or m/s
2. Relative speed direction: Same vs opposite direction
3. Train problems: Including/excluding platform length
4. Average speed: Using arithmetic mean instead of harmonic mean
5. Time calculation: Converting minutes to hours or vice versa

*Master these concepts with regular practice. Focus on speed and accuracy for competitive