Time, Speed, and Distance

Module 05 — Lesson 0005 · In Depth · 35 min

One golden relation governs cars, trains, boats, runners, and escalators: Distance = Speed × Time. Everything else is unit conversion, averaging correctly, or adding/subtracting speeds when two things move.

1. The Golden Formula

Distance = Speed × Time

Derivatives (rearranged, same relation):

Speed = Distance / Time
Time = Distance / Speed

Intuition: Speed is distance per unit time. If you go 60 km in 2 h, speed = 30 km/h. Think "for 1 hour, how far?" → divide distance by time.

2. Unit Conversion — Convert Before You Calculate

Conversion	Multiply by	Remember
`km/h → m/s`	`5/18` (÷3.6)	90 km/h = 25 m/s, 36→10, 72→20
`m/s → km/h`	`18/5` (×3.6)	20 m/s = 72 km/h

Derivation: 1 km =1000m, 1h=3600s → 1 km/h =1000/3600=5/18 m/s. So 90×5/18=25. If a train length is in meters and speed in km/h, convert speed to m/s before dividing.

3. Average Speed — Total Distance / Total Time

Warning: average speed is NOT (S₁+S₂)/2 — that arithmetic mean ignores time weighting.

Average Speed = Total Distance / Total Time

Special Case — Equal Distances

Go A→B at speed x, return B→A at y (equal distances):

Average Speed = (2 × x × y) / (x + y) — harmonic mean of x, y.

Why harmonic? You spend more time at slower speed, so it weighs heavier. Going 40 and 60 over equal distance: arithmetic mean 50 is wrong; actual = 2×40×60/100=48. Derivation: total 2d, time = d/x + d/y → avg = 2/(1/x+1/y).
For unequal distances, fall back to total distance / total time.

4. Relative Speed — Two Objects Moving

Situation	Formula	Use
Same direction	`Srel = Sfast − Sslow`	Overtaking — faster must close the gap
Opposite direction	`Srel = S₁ + S₂`	Meeting / collision — they rush toward each other

Meeting time: Time to meet = Initial Separation / Relative Speed

Worked: Two runners 100m apart, 10 m/s and 6 m/s toward each other → Srel=16, time=100/16=6.25s. Same direction, faster behind 100m → Srel=4, time=25s to catch.

5. Trains and Platforms (Length Matters)

Crossing a pole/man (point): distance = LT (train length)
Crossing a platform/bridge/tunnel: distance = LT + LP
Crossing another train: distance = LT₁ + LT₂, speed = Srel (add if opposite, subtract if same)

Full SVGs and worked times are in Module 06 — the principle here is tail must clear.

6. Boats and Streams — Water Pushes

B: boat in still water
C: current speed
Downstream (with flow): D = B + C
Upstream (against flow): U = B − C
Given D and U: B = (D+U)/2, C = (D−U)/2 — average and half-difference.

Same as Module 06 — here just the definitions for speed problems.

7. Advanced Concepts

A. Races and Circular Tracks

Head start: A gives B a start of x meters → A runs full D, B runs D−x (same time for dead heat). Speed ratio = D/(D−x).
Circular track — first meeting: Time = Circumference / Srel. Same direction: Srel = S₁−S₂ (faster gains one lap); opposite: S₁+S₂. (Full details in Module 24.)

B. Late / Early Concept (Saving Time)

If increasing speed by Δ saves T minutes over the same distance D:

(D / Old_Speed) − (D / New_Speed) = Time_Difference

Worked: Going 40→60 km/h saves 30 min (0.5h) over distance D → D/40 − D/60 =0.5 → D(1/40−1/60)=0.5 → D×(1/120)=0.5 → D=60 km.

C. Escalators / Walkways — Same as Boats

Walking with escalator: Speed = Man + Escalator
Walking against escalator: Speed = Man − Escalator
Stationary (standing) → speed = escalator alone.

Formally identical to downstream/upstream — the medium moves.

8. Summary of Variables

Symbol	Meaning
`D`	Distance (m, km)
`S, v`	Speed (m/s, km/h)
`t`	Time (s, h)
`u`	Initial velocity (acceleration problems)
`a`	Acceleration (m/s²)
`L`	Length (train, tunnel)

Golden formula?

90 km/h in m/s?

Equal distances at x and y → avg speed?

Same direction, Srel=?

Train crossing platform → distance?

Boats: Downstream D and Upstream U → B=?

Saving time: increasing speed saves T. Equation?

Walking with escalator → speed?

Notes

TSD :- Golden :-

D ⇒ S×T ; S ⇒ D/T ; T ⇒ D/S

Units ⇒ km/h→m/s ×5/18 ; m/s→km/h ×18/5

Average & Relative :-

Avg ⇒ total D / total T ; equal D ⇒ 2xy/(x+y) (harmonic)

Srel ⇒ same: Sfast−Sslow ; opposite: S₁+S₂ ; meet: D/Srel

Trains / Boats :-

Train: pole→LT ; platform→LT+LP ; two trains→LT₁+LT₂ with Srel

Boats: D=B+C, U=B−C ; B=(D+U)/2, C=(D−U)/2 ; escalator same

Advanced :-

Race ⇒ head x→A D, B D−x ; circular meet→circum/Srel

Late/early ⇒ D/Sold − D/Snew = ΔT

Primary source: none pinned yet — drop your preferred video/book resource into RESOURCES.md and it will be linked here. Ask me anything that's unclear.

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