Time, Distance & Average

Key Concepts:

• Speed (S) — चाल: वह दर जिस पर कोई वस्तु distance तय करती है।
• Distance (D) — दूरी: तय किया गया कुल रास्ता।
• Time (T) — समय: distance तय करने में लगा समय।

Example: A car travels 240 km in 4 hours. What is the speed?

• Speed = 240/4 = 60 km/h

Example 2: A train travels at 50 km/h for 3 hours. What distance does it cover?

• Distance = 50 × 3 = 150 km

• km/h to m/s: Multiply by 5/18 (or divide by 3.6)
• m/s to km/h: Multiply by 18/5 (or multiply by 3.6)

Example:

• 72 km/h = 72 × 5/18 = 20 m/s
• 25 m/s = 25 × 18/5 = 90 km/h

Other Conversions:

• 1 km = 1000 m
• 1 hour = 60 minutes = 3600 seconds
• 1 km/h = 1000m/3600s = 5/18 m/s

Relative Speed दो moving objects के बीच की speed है।

• Same Direction (समान दिशा): Relative Speed = S₁ − S₂ (S₁ > S₂)
• Opposite Direction (विपरीत दिशा): Relative Speed = S₁ + S₂

Example 1: Two trains moving in the same direction at 60 km/h and 40 km/h.

• Relative Speed = 60 − 40 = 20 km/h

Example 2: Two trains moving in opposite directions at 60 km/h and 40 km/h.

• Relative Speed = 60 + 40 = 100 km/h

Application: Used in problems involving trains crossing each other, overtaking, etc.

Special Case (Same Distance):

Where x and y are the speeds for the two equal distances.

Example 1: A car travels 60 km at 30 km/h and 60 km at 60 km/h. Find average speed.

• Time1 = 60/30 = 2 hours, Time2 = 60/60 = 1 hour
• Total Distance = 120 km, Total Time = 3 hours
• Average Speed = 120/3 = 40 km/h
• Formula Check: 2×30×60/(30+60) = 3600/90 = 40 km/h ✓

Example 2: A person travels at 40 km/h for 2 hours and 60 km/h for 3 hours. Find average speed.

• Total Distance = (40×2) + (60×3) = 80 + 180 = 260 km
• Total Time = 2 + 3 = 5 hours
• Average Speed = 260/5 = 52 km/h

Average (Mean) numbers के set का central value होता है।

Example: Find the average of 5, 10, 15, 20, 25.

• Sum = 5+10+15+20+25 = 75
• Number = 5
• Average = 75/5 = 15

Key Concept:

• Sum = Average × Number of items
• Number of items = Sum / Average

• Average of n numbers: (Sum of n numbers) / n
• When a new number is added: New Average = (Old Sum + New Number) / (n+1)
• When a number is replaced: New Sum = Old Sum − Removed + Added
• Average of first n natural numbers: (n+1)/2
• Average of first n even numbers: (n+1)
• Average of first n odd numbers: n

Example: Average of 5 numbers is 20. If a new number 30 is added, what is the new average?

• Old Sum = 5 × 20 = 100
• New Sum = 100 + 30 = 130
• New Average = 130/6 = 21.67

Weighted Average तब use होता है जब different items के अलग-अलग weights (importance) होते हैं।

Where w = weight, x = value

Example: A student scores 80 in Maths (weight 5), 70 in Science (weight 3), and 90 in English (weight 2). Find weighted average.

• Σ(w × x) = (5×80) + (3×70) + (2×90) = 400 + 210 + 180 = 790
• Σw = 5 + 3 + 2 = 10
• Weighted Average = 790/10 = 79

Application: Used in CGPA, SGPA, combination of groups with different sizes.

• Train crossing a pole/person: Distance = Length of train
• Train crossing a platform: Distance = Length of train + Length of platform
• Train crossing another train (same direction): Distance = Sum of lengths, Relative speed = S₁ − S₂
• Train crossing another train (opposite direction): Distance = Sum of lengths, Relative speed = S₁ + S₂

Example: A train 200 m long, speed 72 km/h (20 m/s). How long to cross a 300 m platform?

• Total distance = 200 + 300 = 500 m
• Time = 500/20 = 25 seconds

• Speed of boat in still water (u): स्थिर पानी में नाव की चाल
• Speed of stream (v): धारा की चाल
• Downstream Speed (with the stream): u + v
• Upstream Speed (against the stream): u − v

Example: A boat goes downstream at 12 km/h and upstream at 8 km/h. Find speed in still water and stream speed.

• u = (12+8)/2 = 10 km/h
• v = (12−8)/2 = 2 km/h

• Speed, Distance, Time: S = D/T, D = S×T, T = D/S
• Unit Conversion: km/h × 5/18 = m/s
• Relative Speed (same direction): S₁ − S₂
• Relative Speed (opposite direction): S₁ + S₂
• Average Speed: Total D / Total T
• Average Speed (equal distances): 2xy/(x+y)
• Average (Simple): Sum / n
• Weighted Average: Σ(w×x) / Σw
• Train crossing platform: Time = (L_train + L_platform) / S
• Boats: Down = u+v, Up = u−v; u = (Down+Up)/2, v = (Down−Up)/2

• Trap 1: Average speed = average of speeds → ❌ Average speed = Total D / Total T (not simple average of speeds).
• Trap 2: Unit conversion — 72 km/h = 20 m/s (not 72 m/s).
• Trap 3: Relative speed in same direction = S₁ − S₂ (not S₁ + S₂).
• Trap 4: Train crossing platform: Distance = Train length + Platform length (not just platform length).
• Trap 5: Downstream = u + v (not u − v).
• Trap 6: Weighted average = simple average when weights are different → ❌ Use weighted average formula.

A car travels 300 km in 5 hours. What is its speed?

• A. 50 km/h
• B. 60 km/h
• C. 70 km/h
• D. 80 km/h

Explanation: Speed = 300/5 = 60 km/h.

Convert 90 km/h to m/s.

• A. 15 m/s
• B. 20 m/s
• C. 25 m/s
• D. 30 m/s

Explanation: 90 × 5/18 = 25 m/s.

Two trains are moving in opposite directions at 60 km/h and 40 km/h. What is their relative speed?

• A. 20 km/h
• B. 60 km/h
• C. 100 km/h
• D. 120 km/h

Explanation: Opposite direction → Relative speed = 60 + 40 = 100 km/h.

A person travels 120 km at 40 km/h and 120 km at 60 km/h. What is the average speed?

• A. 48 km/h
• B. 50 km/h
• C. 52 km/h
• D. 55 km/h

Explanation: Equal distances → Avg = 2×40×60/(40+60) = 4800/100 = 48 km/h.

A train 200 m long, speed 72 km/h (20 m/s). How long to cross a 300 m platform?

• A. 10s
• B. 15s
• C. 20s
• D. 25s

Explanation: Total distance = 200+300 = 500 m; Time = 500/20 = 25 seconds.

A class has 20 boys with average marks 70 and 30 girls with average marks 80. Find the overall average.

• A. 74
• B. 75
• C. 76
• D. 77

Explanation: Weighted avg = (20×70 + 30×80)/(20+30) = (1400+2400)/50 = 3800/50 = 76.

A boat goes downstream at 14 km/h and upstream at 6 km/h. Find speed of stream.

• A. 2 km/h
• B. 4 km/h
• C. 6 km/h
• D. 8 km/h

Explanation: v = (Downstream − Upstream)/2 = (14−6)/2 = 4 km/h.

Key Formulae:

• S = D/T, D = S×T, T = D/S
• km/h → m/s: × 5/18
• Relative Speed: Same direction → S₁−S₂; Opposite → S₁+S₂
• Avg Speed = Total D/Total T; Equal distances → 2xy/(x+y)

Trains:

• Pole/Person: Distance = Train length
• Platform: Distance = Train + Platform
• Other train: Distance = Sum of lengths

Boats & Streams:

• Downstream = u+v, Upstream = u−v
• u = (Down+Up)/2, v = (Down−Up)/2

Exam Formula:

UGC NET में Time, Distance & Average के questions formula-based और application-based होते हैं। Unit conversions और relative speed पर विशेष ध्यान दें।