🚄 Time, Speed & Distance - Complete Theory

Master motion problems - from trains to boats to races!

🎯 Basic Formula

The Golden Triangle

        Distance
         /    \
        /      \
       /        \
    Speed  ×  Time

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

Units:

• Distance: km, m, cm
• Speed: km/hr, m/s, km/min
• Time: hours, minutes, seconds

📐 Unit Conversions

km/hr to m/s

km/hr to m/s: Multiply by 5/18

Example: 72 km/hr = 72 × 5/18 = 20 m/s

m/s to km/hr

m/s to km/hr: Multiply by 18/5

Example: 10 m/s = 10 × 18/5 = 36 km/hr

Quick Memory Trick:

km/hr → m/s: Divide by 3.6 (or × 5/18)
m/s → km/hr: Multiply by 3.6 (or × 18/5)

🔄 Relative Speed

Same Direction

Relative Speed = |Speed₁ - Speed₂|

If A (60 km/hr) and B (40 km/hr) move same direction:
Relative speed = 60 - 40 = 20 km/hr

Opposite Direction

Relative Speed = Speed₁ + Speed₂

If A (60 km/hr) and B (40 km/hr) move opposite:
Relative speed = 60 + 40 = 100 km/hr

🚂 Train Problems

Train Crossing a Pole/Person

Time = Length of Train / Speed of Train

A 100m train at 36 km/hr crosses a pole:
Speed = 36 × 5/18 = 10 m/s
Time = 100/10 = 10 seconds

Train Crossing a Platform/Bridge

Distance = Length of Train + Length of Platform
Time = (Train Length + Platform Length) / Speed

150m train crosses 250m platform at 20 m/s:
Time = (150 + 250) / 20 = 20 seconds

Two Trains Crossing Each Other

Opposite Direction:

Time = (Train₁ Length + Train₂ Length) / (Speed₁ + Speed₂)

Train A: 100m at 20 m/s
Train B: 150m at 30 m/s (opposite direction)
Time = (100 + 150) / (20 + 30) = 250/50 = 5 seconds

Same Direction:

Time = (Train₁ Length + Train₂ Length) / |Speed₁ - Speed₂|

Train A: 100m at 30 m/s
Train B: 150m at 20 m/s (same direction, A overtaking B)
Time = (100 + 150) / (30 - 20) = 250/10 = 25 seconds

🚤 Boats & Streams

Key Concepts

Boat's speed in still water = b km/hr
Stream speed = s km/hr

Downstream speed = b + s  (with current)
Upstream speed = b - s  (against current)

Finding Speeds

If downstream = d and upstream = u:

Boat speed (b) = (d + u) / 2
Stream speed (s) = (d - u) / 2

💡 Solved Examples

Example 1: Basic Speed

Q: A car covers 300 km in 5 hours. Find speed.

Solution:

Speed = Distance / Time
      = 300 / 5
      = 60 km/hr

Answer: 60 km/hr

Example 2: Unit Conversion

Q: Convert 90 km/hr to m/s.

Solution:

90 km/hr = 90 × 5/18 = 25 m/s

Answer: 25 m/s

Example 3: Train Crossing Pole

Q: A 200m long train crosses a pole in 10 seconds. Find speed.

Solution:

Speed = Distance / Time
      = 200 / 10
      = 20 m/s
      = 20 × 18/5 = 72 km/hr

Answer: 72 km/hr

Example 4: Train Crossing Platform

Q: 150m train crosses 350m platform in 25 seconds. Find speed.

Solution:

Total distance = 150 + 350 = 500m
Speed = 500 / 25 = 20 m/s = 72 km/hr

Answer: 72 km/hr

Example 5: Two Trains (Opposite)

Q: Train A (120m) at 60 km/hr meets Train B (180m) at 40 km/hr coming from opposite direction. Time to cross?

Solution:

Convert speeds to m/s:
Train A: 60 × 5/18 = 50/3 m/s
Train B: 40 × 5/18 = 100/9 m/s

Relative speed = 50/3 + 100/9 = 250/9 m/s
Total distance = 120 + 180 = 300m

Time = 300 / (250/9) = 300 × 9/250 = 10.8 seconds

Answer: 10.8 seconds

Example 6: Boat & Stream

Q: Boat speed in still water = 15 km/hr, stream = 3 km/hr. Time to go 36 km downstream?

Solution:

Downstream speed = 15 + 3 = 18 km/hr
Time = Distance / Speed = 36 / 18 = 2 hours

Answer: 2 hours

Example 7: Finding Boat & Stream Speed

Q: Downstream 24 km in 2 hours, upstream 16 km in 2 hours. Find boat and stream speeds.

Solution:

Downstream speed = 24/2 = 12 km/hr
Upstream speed = 16/2 = 8 km/hr

Boat speed = (12 + 8) / 2 = 10 km/hr
Stream speed = (12 - 8) / 2 = 2 km/hr

Answer: Boat = 10 km/hr, Stream = 2 km/hr

Example 8: Relative Speed (Meeting)

Q: Two cars 300 km apart move towards each other at 40 km/hr and 50 km/hr. When will they meet?

Solution:

Relative speed = 40 + 50 = 90 km/hr (opposite direction)
Time = Distance / Speed = 300 / 90 = 3.33 hours

Answer: 3 hours 20 minutes

Example 9: Average Speed

Q: A person travels 100 km at 50 km/hr and returns at 40 km/hr. Find average speed.

Solution:

⚠️ Average speed ≠ (50 + 40) / 2 ✗

Average Speed = Total Distance / Total Time

Time₁ = 100/50 = 2 hours
Time₂ = 100/40 = 2.5 hours
Total time = 4.5 hours

Average speed = 200 / 4.5 = 44.44 km/hr

Shortcut for Equal Distances:

Average Speed = (2 × s₁ × s₂) / (s₁ + s₂)
              = (2 × 50 × 40) / (50 + 40)
              = 4000 / 90
              = 44.44 km/hr ✓

Answer: 44.44 km/hr

🏃 Race Problems

Formula

If A gives B a start of 'x' meters in a race of 'D' meters:

When A finishes D meters, B finishes (D - x) meters

A's speed : B's speed = D : (D - x)

Time-based Start

If A gives B a start of 't' seconds:

A starts 't' seconds after B

Example: 100m race, A gives 5 sec start
B runs for extra 5 seconds before A starts

Example 10: Race with Start

Q: In a 100m race, A beats B by 10m. Find ratio of speeds.

Solution:

When A finishes 100m, B finishes 90m

Speed ratio A:B = 100:90 = 10:9

Answer: 10:9

📊 Important Patterns

Pattern 1: Meeting Point

Two objects start from two points towards each other:
They meet at a point where distance covered is in ratio of their speeds

If A (40 km/hr) and B (60 km/hr) are 100 km apart:
Meeting point from A's side = (40/100) × 100 = 40 km

Pattern 2: Circular Track (Same Direction)

Time to meet again = Circumference / (Speed₁ - Speed₂)

Two runners on 400m track at 5 m/s and 4 m/s:
Meet again after = 400 / (5-4) = 400 seconds

Pattern 3: Circular Track (Opposite Direction)

Time to meet = Circumference / (Speed₁ + Speed₂)

Same track, opposite direction:
Meet after = 400 / (5+4) = 44.44 seconds

⚡ Quick Shortcuts

Shortcut 1: Common Speed Conversions

18 km/hr = 5 m/s
36 km/hr = 10 m/s
54 km/hr = 15 m/s
72 km/hr = 20 m/s
90 km/hr = 25 m/s
108 km/hr = 30 m/s

Shortcut 2: Train Length Quick Check

If train crosses itself (its own length) in 't' seconds:
Length = Speed × Time

Shortcut 3: Distance Ratio

If time is constant:
Distance₁ : Distance₂ = Speed₁ : Speed₂

If speed is constant:
Distance₁ : Distance₂ = Time₁ : Time₂

Shortcut 4: Boat Problems

If downstream and upstream speeds are equal (rare!):
Stream speed = 0 (still water)

If upstream impossible (negative):
Stream speed > Boat speed (boat can't go upstream!)

⚠️ Common Mistakes

❌ Mistake 1: Average Speed

Wrong: Average = (Speed₁ + Speed₂) / 2 ✗
Right: Total Distance / Total Time ✓
(Or use 2s₁s₂/(s₁+s₂) for equal distances)

❌ Mistake 2: Train Length

Wrong: When crossing pole, distance = platform length ✗
Right: Distance = train length only ✓
When crossing platform: Distance = train + platform ✓

❌ Mistake 3: Relative Speed Direction

Wrong: Always adding speeds ✗
Right:
  - Opposite direction → Add
  - Same direction → Subtract ✓

❌ Mistake 4: Unit Mismatch

Wrong: Using km/hr with meters ✗
Right: Convert to same units first ✓

❌ Mistake 5: Boat Speed

Wrong: Downstream = b - s ✗
Right: Downstream = b + s (with current) ✓
       Upstream = b - s (against current) ✓

📝 Practice Problems

Level 1:

1. A car travels 240 km in 4 hours. Find speed.
2. Convert 108 km/hr to m/s.
3. A 150m train crosses a pole in 15 seconds. Find speed in km/hr.

Level 2:

4. A 200m train crosses a 300m bridge in 25 seconds. Find speed.
5. Boat goes 48 km downstream in 3 hours, 36 km upstream in 3 hours. Find boat and stream speeds.
6. Two trains (100m and 150m) at 60 km/hr and 40 km/hr meet head-on. Time to cross?

Level 3:

7. A person travels 60 km at 30 km/hr and 60 km at 20 km/hr. Find average speed.
8. In a 400m race, A beats B by 40m. If A’s speed is 10 m/s, find B’s speed.
9. Two cars start towards each other from points 500 km apart at 50 km/hr and 60 km/hr. After how long and where do they meet?

🔗 Related Topics

Prerequisites:

• Percentage - Speed increase/decrease
• Ratio & Proportion - Speed ratios
• Average - Average speed concept

Related:

• Time & Work - Similar concept with efficiency

Practice:

• Time Speed Distance Questions
• Formula Sheet

Master Time-Speed-Distance - Think in ratios and conversions! 🚄