7  Time, Speed and Distance

7.1 Introduction

The time–speed–distance (TSD) relationship is one of the most important foundations of aptitude tests.
- It connects motion, travel, and average speed.
- It also extends to boats & streams, trains, relative speed, circular tracks, and clocks.

We will develop formulas, shortcuts, traps, solved examples, and layered practice sets.

7.2 1) Fundamental Relationship

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}}, \quad \text{Distance} = \text{Speed} \times \text{Time}, \quad \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

  • Units must be consistent (km/hr with km, m/s with m).
  • Conversion: \(1\ \text{km/hr} = \tfrac{5}{18}\ \text{m/s}\), and \(1\ \text{m/s} = \tfrac{18}{5}\ \text{km/hr}\).

7.3 2) Average Speed

If equal distance covered at speeds \(x\) and \(y\):
\[ \text{Average Speed} = \frac{2xy}{x+y} \]

If distances differ, average speed = \(\frac{\text{Total Distance}}{\text{Total Time}}\).

7.4 3) Relative Speed

  • Same direction: Relative speed = difference of speeds.
  • Opposite direction: Relative speed = sum of speeds.

Example: Two trains at 60 km/h and 40 km/h, opposite directions. Relative speed=100 km/h.

7.5 4) Trains and Platforms

  • Time to pass a pole = \(\tfrac{\text{Length of train}}{\text{Speed}}\).
  • Time to cross platform = \(\tfrac{\text{Length of train + platform}}{\text{Speed}}\).
  • When two trains cross each other: use combined length and relative speed.

7.6 5) Boats and Streams

  • Downstream speed = \(u+v\) (where \(u\)=boat speed in still water, \(v\)=stream speed).
  • Upstream speed = \(u-v\).
  • Effective speed depends on direction.

Example: Boat speed=10 km/h, stream=2 km/h.
Downstream=12, Upstream=8.

7.7 6) Circular Tracks and Meetings

Two people move in a circle of length \(L\):
- Same direction: time to meet = \(L/|\Delta v|\).
- Opposite direction: time to meet = \(L/(v_1+v_2)\).

7.8 7) Clocks (Angle Problems)

  • Angle between hour and minute hand at time \(h:m\):
    \[ \theta = |30h - 5.5m| \]
  • Hands coincide every 65 5/11 minutes.
  • Hands opposite every 30 minutes approx.

7.9 8) Solved Examples

  1. A car travels 120 km in 2 hours. Speed=60 km/h.
  2. Speed of a man = 12 km/h. Find time for 600 m. Convert=12×1000/3600=3.33 m/s. Time=600/3.33=180 s=3 min.
  3. A man covers 100 km at 50 km/h and returns at 25 km/h. Avg speed=2×50×25/(75)=33.33 km/h.
  4. Train 120 m long, speed 54 km/h, crosses platform 180 m. Time? Speed=15 m/s, distance=300 m → time=20 s.
  5. Boat speed 10, stream 4. Find time for 24 km downstream, 16 km upstream.
    Down=24/14=1.71 h, Up=16/6=2.67 h, total≈4.38 h.
  6. Two runners in 400 m track, speeds 6 m/s and 5 m/s, same direction. Meet after 400/(6−5)=400 s.
  7. Angle at 3:15 = |30×3−5.5×15|=|90−82.5|=7.5°.

7.10 9) Common Traps

  • Mixing up relative speed signs.
  • Forgetting to convert units (km/hr ↔︎ m/s).
  • Using harmonic mean only when equal distance.
  • Confusing crossing time with meeting time.
  • Boats: always use \(u+v\) (down) and \(u−v\) (up).
  • Clock angles: formula must be absolute difference.

7.11 Practice Set – Level 1

  1. Distance covered at 60 km/h for 2 h.
  2. Convert 72 km/h into m/s.
  3. A train 100 m long at 36 km/h passes a pole. Time?
  4. Average speed for equal distances at 20 and 30 km/h.
  5. A man walks 5 km in 1 h. Find speed in m/s.

7.12 Practice Set – Level 2

  1. Two trains 120 m and 180 m long at 54 and 72 km/h cross each other opposite directions. Time?
  2. Boat speed 15 km/h, stream 5 km/h. Find time to go 20 km downstream and return.
  3. Circular track 500 m. Two men speeds 5 and 10 m/s opposite directions. Time to meet?
  4. A clock shows 4:20. Find angle between hands.
  5. A man cycles 40 km at 20 km/h and 60 km at 30 km/h. Avg speed?

7.13 Practice Set – Level 3

  1. A train 300 m long at 90 km/h crosses a platform in 30 s. Find platform length.
  2. Two trains length 150 m, 100 m, speeds 45, 36 km/h, same direction. Time to cross?
  3. A swimmer swims 1 km downstream in 10 min, upstream in 15 min. Find speed of swimmer and stream.
  4. At what time between 7 and 8 will hands of clock coincide?
  5. Two cyclists start at same point, speeds 12 and 15 km/h, same direction on circular 300 m track. After how many minutes meet again at start?

7.14 Answer Key (Outline)

Level 1: 1) 120 km; 2) 20 m/s; 3) 10 s; 4) 24 km/h; 5) 1.39 m/s.
Level 2: 6) 10 s; 7) 4 h; 8) 33.3 s; 9) 10°; 10) 25 km/h.
Level 3: 11) 450 m; 12) 100 s; 13) 4,2 km/h; 14) 38 2/11 min; 15) 6 min.

7.15 Summary

  • TSD core: \(S=D/T\).
  • Avg speed: harmonic mean when equal distances.
  • Relative speed: add for opposite, subtract for same.
  • Trains: crossing time = (sum of lengths)/(relative speed).
  • Boats: downstream = \(u+v\), upstream = \(u−v\).
  • Circular tracks use relative speed.
  • Clock angle = |30h−5.5m|.
  • Always ensure unit consistency.