UGC NET Paper 1 Unit 5 Topic 3: Mathematical Aptitude - Time, Speed & Distance

⏱️ Time, Speed & Distance

Master the core concepts of motion mathematics with formulas, shortcut techniques, and previous year UGC NET questions. Covers relative speed, average speed, unit conversions, and train problems.

🔑 Key Formula: Speed = Distance ÷ Time
The fundamental triangle: S×T=D | D÷S=T | D÷T=S

1. Core Concepts

Concept	Formula	Example
Speed	Distance ÷ Time	240 km in 4h → 60 km/h
Distance	Speed × Time	75 km/h × 2h = 150 km
Time	Distance ÷ Speed	300 km ÷ 50 km/h = 6h
Average Speed	Total Distance ÷ Total Time	60km in 1h + 40km in 2h = 100/3 ≈ 33.33 km/h

⚠️ Important Note: Average speed is NOT the arithmetic mean of speeds when time periods differ.
For equal distances: Avg Speed = 2ab/(a+b) [Harmonic Mean]
Example: 60 km/h going, 40 km/h returning → (2×60×40)/(60+40) = 48 km/h

2. Unit Conversions

Essential Conversions:

• 1 km/h = (5/18) m/s
• 1 m/s = (18/5) km/h
• Shortcut: To convert km/h to m/s → multiply by 5/18
• Shortcut: To convert m/s to km/h → multiply by 18/5

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

3. Relative Speed

When two objects move in relation to each other:

Scenario	Formula	Example
Same Direction	|Speed1 - Speed2|	Car A: 60 km/h, Car B: 50 km/h → Relative speed = 10 km/h
Opposite Direction	Speed1 + Speed2	Two trains (70 km/h & 50 km/h) → Relative speed = 120 km/h

Time to Cross/Overtake:
For objects of length L₁ and L₂:
Time = (L₁ + L₂) / Relative Speed
Example: 200m train @60 km/h overtaking 100m train @40 km/h:
Relative speed = 60-40 = 20 km/h = 20×(5/18) m/s
Time = (200+100) / (20×5/18) = 300×18/100 = 54 seconds

4. Train Problems

Key Formulas:

• Train passing pole/post: Time = Length of Train / Speed
• Train passing platform/bridge: Time = (Length of Train + Length of Platform) / Speed
• Two trains crossing: Time = (L₁ + L₂) / (S₁ ± S₂)

Problem Type	Solution Approach
Train crosses pole in 15 sec @54 km/h	Length = Speed×Time = 54×(5/18)×15 = 225m
150m train crosses 250m platform in 20 sec	Speed = (150+250)/20 = 20 m/s = 20×(18/5) = 72 km/h
Two trains (100m & 120m) moving same direction @50 & 30 km/h	Time = (100+120)/(50-30) = 220/20 = 11h (convert speeds to same unit first)

5. Boat & Stream Problems

Key Terms:

• Downstream Speed = Boat Speed + Stream Speed
• Upstream Speed = Boat Speed - Stream Speed
• Boat Speed = (Downstream + Upstream)/2
• Stream Speed = (Downstream - Upstream)/2

Given	Find	Solution
Downstream: 18 km/h Upstream: 12 km/h	Boat & Stream Speed	Boat = (18+12)/2 = 15 km/h Stream = (18-12)/2 = 3 km/h
Boat: 10 km/h Stream: 2 km/h	Time for 24km downstream	Downstream speed = 10+2 = 12 km/h Time = 24/12 = 2h

🔥 Previous Year UGC NET Questions:

1. A train 150m long passes a pole in 5s. Speed? (Ans: 150/5=30 m/s = 30×18/5=108 km/h)
2. Car travels 1/3 distance @60 km/h, rest @30 km/h. Average speed? (Ans: Total D=3d, T=(d/60 + 2d/30), Avg=3d/(5d/60)=36 km/h)
3. Two trains (100m & 200m) @54 & 72 km/h meet. Time to cross? (Ans: Relative speed=54+72=126 km/h=35 m/s, Time=(100+200)/35≈8.57s)
4. Boat goes 15km upstream in 3h. If stream speed=3 km/h, downstream speed? (Ans: Upstream speed=15/3=5 km/h, Boat=5+3=8 km/h, Downstream=8+3=11 km/h)

🚀 Problem Solving Strategies

Step-by-Step Approach:

1. Identify given values (convert units if needed)
2. Determine what needs to be found
3. Choose appropriate formula
4. Calculate carefully (watch unit consistency)
5. Verify using approximation

Common Mistakes to Avoid:

• Mixing km/h and m/s without conversion
• Using arithmetic mean for average speed when times differ
• Ignoring lengths of objects in relative speed problems

⏱️ Time-Saving Tips:

• Memorize common conversions (5/18 for km/h↔m/s)
• For trains: Always add lengths when calculating crossing time
• In boats: Remember downstream = boat + stream, upstream = boat - stream