Time, Distance & Speed

Master the concepts of time, distance, and speed with our comprehensive guide. Learn about relative speed, trains, boats, and problem-solving techniques.

Table of Contents

Simplified Quantitative Formulas: Speed, Time & Distance

  • Basic Formula: Distance = Speed × Time. Rearranged: Speed = Distance/Time, Time = Distance/Speed.
  • Unit Conversions: 1 m/s = 3.6 km/h. To convert m/s to km/h, multiply by 3.6; km/h to m/s, multiply by 5/18.
  • Relative Speed: If two objects move towards each other, add speeds; same direction, subtract.
  • Speed Ratio: If speeds are in ratio a:b, times taken are in ratio b:a.
  • Average Speed: Average Speed = Total Distance / Total Time. For equal distances at S₁ and S₂, average = 2S₁S₂/(S₁+S₂).
  • Variable Definitions: S = speed, D = distance, T = time, S₁, S₂ = different speeds, a, b = speed ratios.

What do these mean? (Super Simple Explanations & Examples)

  • Basic Formula: If you travel 100 km in 2 hours, speed = 100/2 = 50 km/h.
  • Unit Conversions: 10 m/s = 36 km/h (10×3.6).
  • Relative Speed: Two trains at 40 km/h and 60 km/h towards each other: relative speed = 100 km/h.
  • Speed Ratio: If A:B = 2:3, then time ratio = 3:2.
  • Average Speed: 60 km at 30 km/h, 60 km at 60 km/h: Avg = 2×30×60/(30+60) = 40 km/h.
  • Variable Definitions: S = speed, D = distance, T = time, S₁, S₂ = different speeds, a, b = speed ratios.

1. Basic Concepts

(a) Understanding Speed

Speed is the rate at which an object covers distance.

Basic Formulas:

Speed = Distance/Time

Distance = Speed × Time

Time = Distance/Speed

Units:

  • Speed: km/h, m/s
  • Distance: km, m
  • Time: h, min, s

Example 1: Basic Speed

Distance = 120 km

Time = 2 hours

Speed = 120/2 = 60 km/h

Speed Units Conversion

From To Multiply by
km/h m/s 5/18
m/s km/h 18/5

(b) Average Speed

Average speed is the total distance divided by total time.

Average Speed Formula:

Average Speed = Total Distance/Total Time

For equal distances:

Average Speed = 2xy/(x+y)

where x and y are speeds

Example 2: Average Speed

Going: 60 km/h

Returning: 40 km/h

Average Speed = 2×60×40/(60+40) = 48 km/h

2. Relative Speed

(a) Same Direction

When objects move in the same direction.

Relative Speed (Same Direction):

Relative Speed = |Speed₁ - Speed₂|

Time to meet = Distance/Relative Speed

Example 3: Same Direction

Speed₁ = 60 km/h

Speed₂ = 40 km/h

Distance = 100 km

Relative Speed = 60-40 = 20 km/h

Time to meet = 100/20 = 5 hours

(b) Opposite Direction

When objects move towards each other.

Relative Speed (Opposite Direction):

Relative Speed = Speed₁ + Speed₂

Time to meet = Distance/Relative Speed

Example 4: Opposite Direction

Speed₁ = 50 km/h

Speed₂ = 30 km/h

Distance = 200 km

Relative Speed = 50+30 = 80 km/h

Time to meet = 200/80 = 2.5 hours

3. Trains

(a) Train Crossing Stationary Objects

When a train crosses a stationary object (pole, person, etc.).

Basic Formula:

Time = Length of Train/Speed of Train

Important Points:

  • Length of stationary object is negligible
  • Time taken is independent of the speed of the stationary object
  • Speed should be in the same unit as length

Example 1: Train Crossing Pole

Train length = 200 m

Speed = 72 km/h = 20 m/s

Time = 200/20 = 10 seconds

Example 2: Train Crossing Man

Train length = 150 m

Speed = 54 km/h = 15 m/s

Time = 150/15 = 10 seconds

(b) Train Crossing Platform/Bridge

When a train crosses a platform or bridge.

Formula:

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

Important Points:

  • Total distance = Train length + Platform length
  • Time taken depends on both lengths
  • Speed should be converted to appropriate units

Example 3: Train Crossing Platform

Train length = 200 m

Platform length = 300 m

Speed = 72 km/h = 20 m/s

Time = (200+300)/20 = 25 seconds

Example 4: Train Crossing Bridge

Train length = 250 m

Bridge length = 750 m

Speed = 90 km/h = 25 m/s

Time = (250+750)/25 = 40 seconds

(c) Trains Crossing Each Other

When two trains cross each other.

Opposite Direction:

Time = (Length₁ + Length₂)/(Speed₁ + Speed₂)

Same Direction:

Time = (Length₁ + Length₂)/|Speed₁ - Speed₂|

Important Points:

  • For opposite direction, speeds are added
  • For same direction, speeds are subtracted
  • Total distance is sum of both train lengths

Example 5: Trains Crossing (Opposite)

Train₁: 200 m, 60 km/h

Train₂: 300 m, 40 km/h

Time = (200+300)/(60+40) = 5 seconds

Example 6: Trains Crossing (Same Direction)

Train₁: 200 m, 60 km/h

Train₂: 300 m, 40 km/h

Time = (200+300)/(60-40) = 25 seconds

(d) Train Overtaking

When one train overtakes another.

Overtaking Formula:

Time = (Length₁ + Length₂)/(Speed₁ - Speed₂)

Important Points:

  • Faster train must be behind initially
  • Time taken is from when front of faster train reaches back of slower train
  • Overtaking is complete when back of faster train passes front of slower train

Example 7: Train Overtaking

Train₁: 200 m, 60 km/h

Train₂: 300 m, 40 km/h

Time = (200+300)/(60-40) = 25 seconds

(e) Train in Tunnel

When a train passes through a tunnel.

Tunnel Formula:

Time = (Length of Train + Length of Tunnel)/Speed of Train

Important Points:

  • Total distance = Train length + Tunnel length
  • Time starts when front of train enters tunnel
  • Time ends when back of train exits tunnel

Example 8: Train in Tunnel

Train length = 200 m

Tunnel length = 800 m

Speed = 72 km/h = 20 m/s

Time = (200+800)/20 = 50 seconds

(f) Train and Moving Object

When a train crosses a moving object.

Moving Object Formula:

Same Direction:

Time = Length of Train/(Speed of Train - Speed of Object)

Opposite Direction:

Time = Length of Train/(Speed of Train + Speed of Object)

Important Points:

  • For same direction, subtract speeds
  • For opposite direction, add speeds
  • Length of moving object is negligible

Example 9: Train and Moving Object

Train length = 200 m

Train speed = 60 km/h

Object speed = 20 km/h

Same direction:

Time = 200/(60-20) = 5 seconds

Opposite direction:

Time = 200/(60+20) = 2.5 seconds

(g) Important Theorems

Key Theorems:

  1. If a train of length L₁ crosses a platform of length L₂ in time T, then:
    Speed = (L₁ + L₂)/T
  2. If two trains of lengths L₁ and L₂ cross each other in time T:
    For opposite direction: Speed₁ + Speed₂ = (L₁ + L₂)/T
    For same direction: |Speed₁ - Speed₂| = (L₁ + L₂)/T
  3. If a train overtakes another train in time T:
    Time = (L₁ + L₂)/(Speed₁ - Speed₂)
  4. If a train crosses a moving object:
    Time = L/(Speed of Train ± Speed of Object)

Example 10: Complex Problem

Train A (200 m) crosses a platform (300 m) in 25 seconds

Train B (300 m) crosses the same platform in 30 seconds

Find the time taken for Train A to overtake Train B

Solution:

Speed of A = (200+300)/25 = 20 m/s

Speed of B = (300+300)/30 = 20 m/s

Time to overtake = (200+300)/(20-20) = Not possible (same speed)

4. Boats & Streams

(a) Downstream & Upstream

Speed of boat in still water and speed of stream.

Boat Speed Formulas:

Downstream Speed = Boat Speed + Stream Speed

Upstream Speed = Boat Speed - Stream Speed

Boat Speed = (Downstream + Upstream)/2

Stream Speed = (Downstream - Upstream)/2

Example 7: Boat Speed

Downstream = 20 km/h

Upstream = 12 km/h

Boat Speed = (20+12)/2 = 16 km/h

Stream Speed = (20-12)/2 = 4 km/h

(b) Time and Distance

Calculating time and distance in boats and streams.

Time and Distance:

Time = Distance/Speed

For round trip:

Average Speed = 2×Downstream×Upstream/(Downstream+Upstream)

Example 8: Round Trip

Distance = 24 km

Downstream = 12 km/h

Upstream = 8 km/h

Time downstream = 24/12 = 2 hours

Time upstream = 24/8 = 3 hours

Total time = 5 hours

5. Advanced Concepts

(a) Circular Motion

Objects moving in circular paths.

Circular Motion:

Time to meet = Circumference/Relative Speed

For same direction:

Time = Circumference/|Speed₁ - Speed₂|

For opposite direction:

Time = Circumference/(Speed₁ + Speed₂)

Example 9: Circular Motion

Circumference = 400 m

Speed₁ = 5 m/s

Speed₂ = 3 m/s

Same direction:

Time = 400/(5-3) = 200 seconds

(b) Escalator Problems

Problems involving escalators and moving walkways.

Escalator Formula:

Effective Speed = Person's Speed ± Escalator Speed

Time = Length/Effective Speed

Example 10: Escalator

Escalator length = 100 m

Person's speed = 2 m/s

Escalator speed = 1 m/s

Time = 100/(2+1) = 33.33 seconds

(c) Important Theorems

Key Theorems:

  1. If two objects start from the same point and move in the same direction, the time to meet is Distance/Relative Speed
  2. If two objects start from the same point and move in opposite directions, the time to meet is Distance/Sum of Speeds
  3. For equal distances, average speed is harmonic mean of speeds
  4. For equal times, average speed is arithmetic mean of speeds

Example 11: Average Speed

Equal distances:

Speeds: 30 km/h, 60 km/h

Average Speed = 2×30×60/(30+60) = 40 km/h

Equal times:

Average Speed = (30+60)/2 = 45 km/h

Practice Questions

Test your understanding of Speed, Time & Distance with 20 fully solved, step-by-step questions designed for beginners.

Start Practice