Unit 4: Time, Speed and Distance

I. Orientation — The Governing Relationship

Motion-based aptitude problems examine how far an object travels, how quickly it moves, and how long the movement takes. Their governing principle is that, for uniform motion, distance equals speed multiplied by time. Most problems are solved by identifying compatible units, selecting the correct effective speed, and applying proportional reasoning.

• Fundamental equation: For constant speed,
  
  TEXT

    d = s × t
    s = d / t
    t = d / s

  
  where d is distance, s is speed, and t is time.
• Standard units:
  • Distance: metre (m) or kilometre (km).
  • Time: second (s), minute (min), or hour (h).
  • Speed: metres per second (m/s) or kilometres per hour (km/h).
• Uniform-motion assumption: The basic equation directly applies when speed remains constant throughout the stated interval.
• Effective-speed principle: In train, relative-motion, and boat problems, s must represent the speed at which the relevant distance is being covered—not necessarily the speed shown for one object alone.
• Unit-consistency rule: Distance and time must use corresponding units before substitution; kilometres pair naturally with hours, while metres pair with seconds.

II. Fundamental Quantities and Calculations — Measuring Motion

The basic quantities of motion are linked mathematically but represent different physical ideas: distance measures path length, time measures duration, and speed measures the rate at which distance is covered.

A. Time, speed and distance

Time, speed, and distance form a three-variable relationship in which any one quantity can be calculated from the other two.

• Distance: Distance is the total path covered and is always non-negative; travelling 60 km east and then 20 km west gives a total distance of 80 km.
• Time: Time is the duration of travel; 2 h 30 min must be written as 2.5 h when speed is in km/h.
• Speed: Speed is the distance covered per unit time; a vehicle covering 150 km in 3 h has speed 50 km/h.
• Formula selection: The unknown quantity determines the form used:
  
  TEXT

    Distance = Speed × Time
    Speed = Distance ÷ Time
    Time = Distance ÷ Speed

• Worked example: A cyclist moving at 18 km/h for 40 min travels:
  
  TEXT

    t = 40/60 h = 2/3 h
    d = 18 × 2/3 = 12 km

B. Concept of time, speed, and distance

The conceptual relationship is based on how one quantity changes when another is fixed.

• Fixed speed: Distance is directly proportional to time:
  
  TEXT

    d ∝ t

  
  At 40 km/h, doubling travel time from 2 h to 4 h doubles distance from 80 km to 160 km.
• Fixed time: Distance is directly proportional to speed; in 3 h, speeds of 20 km/h and 30 km/h produce distances in the ratio 2:3.
• Fixed distance: Speed is inversely proportional to time:
  
  TEXT

    s ∝ 1/t

  
  If speed doubles, the time needed for the same journey is halved.
• Percentage relation: For a fixed distance, increasing speed by x% reduces time by:
  
  TEXT

    Time reduction % = [x / (100 + x)] × 100

  
  Thus, a 25% speed increase reduces travel time by 20%.
• Scalar character: Speed has magnitude but no direction; direction becomes relevant when velocity or relative motion is considered.

C. Conversion of units and proportionality

Unit conversion makes quantities compatible, while proportionality permits rapid comparison without repeatedly calculating absolute values.

• Speed conversion:
  
  TEXT

    1 km/h = 5/18 m/s
    1 m/s = 18/5 km/h

  
  The factor follows from 1 km = 1000 m and 1 h = 3600 s.
• Conversion direction:
  • Multiply km/h by 5/18 to obtain m/s.
  • Multiply m/s by 18/5 to obtain km/h.
• Time conversion:
  
  TEXT

    1 h = 60 min = 3600 s
    1 min = 60 s

  
  A time of 1 h 45 min equals 1.75 h or 6300 s.
• Ratio method: If two objects travel for equal times, their distances are proportional to their speeds:
  
  TEXT

    d₁/d₂ = s₁/s₂

  
  Here d₁ and d₂ are distances, while s₁ and s₂ are corresponding speeds.
• Inverse ratio: For an equal distance,
  
  TEXT

    t₁/t₂ = s₂/s₁

  
  where t₁ and t₂ are corresponding travel times.
• Worked example: A speed of 72 km/h becomes:
  
  TEXT

    72 × 5/18 = 20 m/s

  
  Therefore, the object covers 20 m each second.

D. Average speed concept

Average speed is total distance divided by total time, not generally the arithmetic mean of separate speeds.

• General formula:
  
  TEXT

    Average speed = Total distance / Total time
    s_avg = (d₁ + d₂ + ...)/(t₁ + t₂ + ...)

  
  where s_avg is average speed and each d and t represents a journey segment.
• Equal time intervals: If speeds s₁ and s₂ are maintained for equal durations,
  
  TEXT

    s_avg = (s₁ + s₂)/2

  
  because each speed contributes for the same amount of time.
• Equal distances: If equal distances are covered at speeds s₁ and s₂,
  
  TEXT

    s_avg = 2s₁s₂/(s₁ + s₂)

  
  This harmonic-mean form is less than the arithmetic mean whenever the speeds differ.
• Stops and delays: Waiting time belongs in total time if average speed is measured over the complete journey.
• Worked example: A person travels 60 km at 30 km/h and returns 60 km at 60 km/h:
  
  TEXT

    Total distance = 120 km
    Total time = 60/30 + 60/60 = 3 h
    Average speed = 120/3 = 40 km/h

  
  The answer is not 45 km/h because the travel times are unequal.

III. Relative Motion — Comparing Moving Objects

Relative motion describes how quickly the distance between two moving objects changes. Direction determines whether their speeds are added or subtracted.

A. Relative speed concept and application

Relative speed is the speed of one moving object as observed from another moving object.

• Same direction: The faster object gains on the slower object at the difference of their speeds:
  
  TEXT

    s_rel = |s₁ - s₂|

  
  where s_rel is relative speed and s₁, s₂ are object speeds.
• Opposite directions: The separation closes or increases at the sum of their speeds:
  
  TEXT

    s_rel = s₁ + s₂

• Meeting time: If objects move toward each other from an initial separation D,
  
  TEXT

    t = D/(s₁ + s₂)

  
  where D is initial distance and t is meeting time.
• Overtaking time: For objects moving in the same direction,
  
  TEXT

    t = D/|s₁ - s₂|

  
  provided the faster object is behind and D is their initial separation.
• Worked example: Two vehicles 210 km apart approach each other at 60 km/h and 45 km/h:
  
  TEXT

    Relative speed = 60 + 45 = 105 km/h
    Meeting time = 210/105 = 2 h

IV. Train Motion — Length-Based Relative-Speed Problems

Train problems combine relative speed with the physical lengths of trains, platforms, bridges, or tunnels. The distance used is the total length that must clear the reference object.

A. Problems on trains

A train completely crosses an object only when its rear end has passed that object, so train length must be included in the crossing distance.

• Crossing a point: A pole, person, or signal has negligible length:
  
  TEXT

    t = L/s

  
  where L is train length and s is train speed in compatible units.
• Crossing a platform:
  
  TEXT

    t = (L + P)/s

  
  where P is platform length.
• Two trains, opposite directions:
  
  TEXT

    t = (L₁ + L₂)/(s₁ + s₂)

  
  where L₁ and L₂ are train lengths.
• Two trains, same direction:
  
  TEXT

    t = (L₁ + L₂)/|s₁ - s₂|

  
  The faster train must gain a distance equal to the sum of both lengths.
• Moving observer: A person walking opposite to a train produces added relative speed; walking in the train’s direction produces subtracted relative speed.
• Worked example: A 150 m train moving at 54 km/h crosses a 100 m platform:
  
  TEXT

    Speed = 54 × 5/18 = 15 m/s
    Distance = 150 + 100 = 250 m
    Time = 250/15 = 16⅔ s

V. Water Travel — Motion with and against a Current

Boat problems distinguish the boat’s speed in still water from the stream’s speed. The current assists downstream travel and opposes upstream travel.

A. Boats and streams

The actual speed of a boat relative to the bank is obtained by combining its still-water speed with the stream current.

• Still-water speed: Let b represent the boat’s speed when no current acts.
• Stream speed: Let c represent the current’s speed relative to the bank.
• Direction effect:
  1. With the current: The stream assists the boat, so effective speed increases.
  2. Against the current: The stream resists the boat, so effective speed decreases.
• Recovery formulas: If downstream speed is d and upstream speed is u,
  
  TEXT

    b = (d + u)/2
    c = (d - u)/2

  
  These follow from adding and subtracting the downstream and upstream equations.
• Feasibility condition: Upstream travel requires b > c; if b = c, upstream speed is zero.
• Time relation: For a water-route distance D,
  
  TEXT

    Time = D/Effective speed

  
  Distance must be measured along the route travelled.

B. Downstream and upstream

Downstream and upstream speeds are paired effective speeds created by the same boat and current acting in opposite ways.

• Downstream motion: The boat and stream move in the same direction:
  
  TEXT

    Downstream speed = b + c

• Upstream motion: The boat moves opposite to the stream:
  
  TEXT

    Upstream speed = b - c

• Round-trip time: For equal one-way distance D,
  
  TEXT

    Total time = D/(b + c) + D/(b - c)

  
  where b is still-water speed and c is current speed.
• Average round-trip speed: For equal downstream and upstream distances,
  
  TEXT

    s_avg = 2(b + c)(b - c)/[(b + c) + (b - c)]
          = (b² - c²)/b

  
  This is below b whenever the stream speed is non-zero.
• Worked example: A boat travels at 12 km/h downstream and 8 km/h upstream:
  
  TEXT

    Still-water speed = (12 + 8)/2 = 10 km/h
    Stream speed = (12 - 8)/2 = 2 km/h

  
  Thus, the current adds 2 km/h downstream and subtracts 2 km/h upstream.