Time, Speed, and Distance

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Time, Speed, and Distance — Key Facts for NABE (Pakistan)

• Speed = Distance / Time | Distance = Speed × Time | Time = Distance / Speed
• Average Speed (for equal distances at speeds v₁ and v₂): = 2v₁v₂/(v₁+v₂) — Harmonic Mean
• Relative Speed (opposite directions): Sum of speeds | Same direction: Difference
• Train Problems: When train passes pole/platform, use train’s own length
• ⚡ Exam tip: Always convert units — km/hr to m/s multiply by 5/18; m/s to km/hr multiply by 18/5

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Time, Speed, and Distance — NABE (Pakistan) Study Guide

Basic Formulas

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

Units:

• Speed: km/hr or m/s
• Distance: km or m
• Time: hours or seconds

Unit Conversion

km/hr to m/s:

km/hr × (1000m/3600s) = km/hr × (5/18)
Example: 72 km/hr = 72 × 5/18 = 20 m/s

m/s to km/hr:

m/s × (18/5) = km/hr
Example: 15 m/s = 15 × 18/5 = 54 km/hr

Average Speed

When equal distances are covered at different speeds:

Average Speed = (v₁ + v₂) / 2   [ONLY when distances are equal]

Correct Formula for Equal Distances:

v_avg = 2v₁v₂/(v₁ + v₂)   [Harmonic Mean]

Example: A car travels 100 km at 40 km/hr and returns 100 km at 60 km/hr

• v_avg = 2×40×60/(40+60) = 4800/100 = 48 km/hr

NOT: (40+60)/2 = 50 km/hr (This is incorrect!)

Relative Speed

Two objects moving in opposite directions:

Relative Speed = v₁ + v₂

Two objects moving in same direction:

Relative Speed = |v₁ - v₂|

NABE Exam Pattern

Common question types:

1. Basic speed/distance/time calculations
2. Average speed with equal distances
3. Relative speed (trains, cars)
4. Train passing pole or platform
5. Upstream and downstream problems

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Time, Speed, and Distance — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Speed Fundamentals

Definition: Speed is the rate at which an object covers distance.

Key Characteristics:

• Speed can be constant or variable
• Average speed considers total distance and total time
• Instantaneous speed is speed at a particular moment

Formula Triangle:

    Distance
   /        \
 Speed — Time

Cover: D = S × T, S = D/T, T = D/S

2. Average Speed — Complete Analysis

Case 1: Equal Distances at Different Speeds

• Distance d at speed v₁ takes time d/v₁
• Distance d at speed v₂ takes time d/v₂
• Total distance = 2d, Total time = d/v₁ + d/v₂
• v_avg = 2d/(d/v₁ + d/v₂) = 2v₁v₂/(v₁ + v₂)

Case 2: Equal Times at Different Speeds

• Distance covered in equal time t at v₁ = v₁t
• Distance covered in equal time t at v₂ = v₂t
• Total distance = (v₁ + v₂)t
• v_avg = (v₁ + v₂)t/(2t) = (v₁ + v₂)/2

Important: When speeds are given without specifying distance or time, assume equal distances for average speed problems.

3. Relative Speed — Two Objects

Moving in Same Direction:

Relative Speed = Difference of speeds = |v₁ - v₂|

Problem: Two trains A (72 km/hr) and B (54 km/hr) move in same direction. If initially A is 20 km behind B, find time for A to overtake B.

• Relative speed = 72 - 54 = 18 km/hr
• Distance to cover = 20 km
• Time = 20/18 = 10/9 hours = 1 hour 7 minutes

Moving in Opposite Directions:

Relative Speed = Sum of speeds = v₁ + v₂

Problem: Two trains A (72 km/hr) and B (54 km/hr) approach each other. Initial distance = 126 km. Find time to meet.

• Relative speed = 72 + 54 = 126 km/hr
• Time = 126/126 = 1 hour

4. Train Problems

Train Passing a Pole:

• The train covers its own length
• Distance = Length of train (L)
• Time = L/Speed

Train Passing a Platform:

• The train covers its own length + platform length
• Distance = L_train + L_platform
• Time = (L_train + L_platform)/Speed

Train Passing Another Train:

• Opposite directions: Sum of lengths / Sum of speeds
• Same direction: Sum of lengths / Difference of speeds

Example: A train 150m long passes a pole in 10 seconds. Find speed.

• Distance = 150m, Time = 10s
• Speed = 150/10 = 15 m/s = 15 × 18/5 = 54 km/hr

Example: A train 200m long passes a platform 150m long in 20 seconds. Find speed.

• Distance = 200 + 150 = 350m
• Speed = 350/20 = 17.5 m/s = 63 km/hr

5. Boats and Streams (Upstream/Downstream)

Key Terms:

• Still Water Speed (u): Speed of boat in stationary water
• Stream/Current Speed (v): Speed of water flow
• Downstream Speed: u + v (boat moving with current)
• Upstream Speed: u - v (boat moving against current)

Formulas:

Downstream Speed = u + v
Upstream Speed = u - v

u = (Downstream + Upstream)/2
v = (Downstream - Upstream)/2

Example: A boat travels downstream in 4 hours and upstream in 6 hours. Distance between points = 48 km. Find speed of boat in still water.

• Downstream speed = 48/4 = 12 km/hr
• Upstream speed = 48/6 = 8 km/hr
• Boat speed = (12 + 8)/2 = 10 km/hr
• Stream speed = (12 - 8)/2 = 2 km/hr

6. Circular Motion

When two runners start from same point:

• Same direction: Time = LCM of times or (distance)/(relative speed)
• Opposite directions: Time = (distance)/(sum of speeds)

Example: Two runners on circular track (1 km circumference), speeds 6 m/s and 4 m/s. When do they meet?

• Relative speed = 2 m/s
• Time to meet = 1000/2 = 500 seconds

7. Conversion Tricks

km/hr	m/s
1	5/18
18	5
36	10
54	15
72	20
90	25
108	30

Quick Conversions:

• To convert m/s to km/hr: multiply by 3.6
• To convert km/hr to m/s: divide by 3.6

8. Common Mistakes to Avoid

1. Unit Mixing: Always convert to same units before calculating
2. Average Speed Formula: Don’t use (v₁+v₂)/2 unless times are equal
3. Train Length: Remember to include train’s own length when passing objects
4. Downstream/Upstream: Identify current direction correctly
5. Relative Speed Direction: Add for opposite, subtract for same direction

Practice Questions for NABE

1. A train 150m long is moving at 60 km/hr. How long will it take to pass a pole?
2. Two cities A and B are 300 km apart. One train leaves A at 8 AM at 50 km/hr and another from B at 10 AM at 70 km/hr. When will they meet?
3. A man rows upstream at 12 km/hr and downstream at 18 km/hr. Find speed of stream and speed in still water.
4. A car covers 180 km in 3 hours going and returns in 4 hours. Find average speed.
5. Two trains 150m and 200m long run at 40 km/hr and 30 km/hr respectively in opposite directions. How long to cross each other?

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