Time, Speed and Distance

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What: Time, Speed and Distance covers the basic identity $d = vt$, relative speed (same direction, opposite direction), average speed, train problems (crossing a pole, platform, another train), and boat-and-stream problems.
Why it matters: CDS papers from 2000 to 2023 carry 4–7 questions per sitting on this single chapter — the highest weightage among arithmetic word-problem topics.
Key fact: Speed conversion: 1 km/h = $\tfrac{5}{18}$ m/s. Multiply km/h by 5/18 to get m/s; multiply m/s by 18/5 to get km/h. Memorise — it's used in nearly every train problem.

Time, Speed and Distance is the highest-yield arithmetic chapter in CDS. The patterns are clean, the formulas few, and the marks plentiful. The trick is fluent unit conversion (km/h ↔ m/s), the relative-speed rule for trains and meetings, and the harmonic mean for average speed on equal distances.

This page is built from CDS Previous Year Questions across 2000–2023. Pair with Ratio and Proportion (ratio of speeds = inverse ratio of times for same distance) and Average (average-speed harmonic mean).

What This Topic Covers

CDS scope: (1) basic identity — $d = vt$; (2) unit conversion — km/h ↔ m/s; (3) average speed — harmonic mean for equal distances; (4) relative speed — same direction ($v_1 - v_2$) and opposite direction ($v_1 + v_2$); (5) trains — crossing a pole, platform, another train, and a person; (6) boats and streams — upstream, downstream, still water; (7) circular tracks — meeting again; and (8) two-person meeting / pursuit.

Why This Topic Matters

4–7 CDS questions per paper — the single largest word-problem head in arithmetic.
Conversion fluency is the difference between 30-second and 90-second train problems.
Most archetypes have one-line shortcut formulas (boat: still-water speed = $(u+d)/2$; meeting time on circular track = LCM of lap times).

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-II	4	Basic, train-platform, average speed
2009-II	3	Train-train opposite, boat-stream
2010-I/II	5	Relative speed, circular track
2011-I/II	4	Train-pole, boat upstream
2012-I/II	5	Average speed, two-train crossing
2013-I/II	5	Two-person meeting, train-train
2014-I/II	4	Boat-stream ratio, circular track
2015-I/II	4	Train-platform, average speed
2016-I/II	4	Pursuit, relative speed
2017-I/II	5	Mixed
2018-I/II	4	Boat, train, circular
2019-II	3	Train, average speed
2020-I/II	5	Mixed
2021-I/II	4	Boat-stream, pursuit
2022-I / 2023-I	5	Mixed

⚡ CDS Alert

Two trains in opposite directions cross each other in time $t = (L_1 + L_2)/(v_1 + v_2)$. Same direction: $t = (L_1 + L_2)/|v_1 - v_2|$. Always convert speeds to m/s first if lengths are in metres and time in seconds.

Core Concepts

Basic Identity and Conversion

Distance = Speed × Time $$d = v \cdot t, \quad v = \frac{d}{t}, \quad t = \frac{d}{v}$$

Unit Conversion $$1 \text{ km/h} = \frac{5}{18} \text{ m/s}, \qquad 1 \text{ m/s} = \frac{18}{5} \text{ km/h}$$

Average Speed

Equal Distances, Different Speeds $$\bar{v} = \frac{2 v_1 v_2}{v_1 + v_2} \quad (\text{harmonic mean})$$

Equal Times, Different Speeds $$\bar{v} = \frac{v_1 + v_2}{2} \quad (\text{arithmetic mean})$$

Relative Speed

Same direction: relative speed $= v_1 - v_2$ (faster catches up). Opposite direction: $v_1 + v_2$ (closing speed).

Trains

Time for a train of length $L$ at speed $v$:

Train Crossing Pole (or stationary person): $t = L/v$.
Platform of length $P$: $t = (L + P)/v$.
Another train of length $L_2$, opposite direction at $v_2$: $t = (L + L_2)/(v + v_2)$.
Another train of length $L_2$, same direction at $v_2$: $t = (L + L_2)/|v - v_2|$.

Boats and Streams

Boat Speeds Downstream speed $= u + s$; Upstream speed $= u - s$.
Still-water speed $u = (d + u) / 2$; Stream speed $s = (d - u)/2$.
Where $d = $ downstream and $u = $ upstream observed speeds.

Pursuit and Meeting

Two persons start from the same point in the same direction with speeds $v_1 > v_2$. The faster catches up the slower over relative speed $v_1 - v_2$. If $v_1$ lags by distance $D$, catch-up time = $D / (v_1 - v_2)$.

Circular Track

Two runners on a circular track of length $L$ at speeds $v_1$ and $v_2$:

Meeting on Circular Track Same direction: meet first time after $L / (v_1 - v_2)$.
Opposite direction: meet first time after $L / (v_1 + v_2)$.
Meeting at start: LCM of $L/v_1$ and $L/v_2$.

Worked Examples

Example 1 — Train Crosses Pole (2011-I)

Q: A train 180 m long is moving at 72 km/h. How long does it take to cross a pole?

Convert 72 km/h to m/s: $72 \cdot 5/18 = 20$ m/s.
Time = length / speed = $180/20 = 9$ seconds.

Example 2 — Train Crosses Platform (2007-II)

Q: A train of length 250 m crosses a 350 m platform in 25 seconds. Find its speed in km/h.

Total distance covered = train + platform = $250 + 350 = 600$ m.
Speed = $600/25 = 24$ m/s.
Convert: $24 \cdot 18/5 = 86.4$ km/h.

Example 3 — Average Speed (2012-II)

Q: A man travels first half of a journey at 30 km/h and the second half at 60 km/h. Find his average speed.

Equal distances → harmonic mean: $\bar{v} = 2 \cdot 30 \cdot 60 / (30 + 60) = 3600/90 = 40$ km/h.

Example 4 — Two Trains Opposite Direction (2013-II)

Q: Two trains of length 120 m and 80 m travel toward each other at 60 km/h and 30 km/h. How long do they take to cross each other?

Relative speed = $60 + 30 = 90$ km/h $= 90 \cdot 5/18 = 25$ m/s.
Combined length = $120 + 80 = 200$ m. Time = $200/25 = 8$ seconds.

Example 5 — Boat Upstream and Downstream (2014-I)

Q: A boat goes 30 km downstream in 3 h and 18 km upstream in 3 h. Find the speed in still water and stream.

Downstream speed = $30/3 = 10$ km/h. Upstream speed = $18/3 = 6$ km/h.
Still water = $(10 + 6)/2 = 8$ km/h. Stream = $(10 - 6)/2 = 2$ km/h.

Example 6 — Pursuit (2016-I)

Q: A walks at 4 km/h and B at 6 km/h. B starts 30 minutes after A. When will B overtake A?

A has a 30-min head start, covering $4 \cdot 0.5 = 2$ km when B starts.
Relative speed = $6 - 4 = 2$ km/h. B closes 2 km in $2/2 = 1$ hour.
B overtakes 1 hour after starting, i.e. 1.5 hours after A started.

Example 7 — Circular Track (2018-I)

Q: Two runners start from the same point on a 400 m circular track running at 4 m/s and 6 m/s in the same direction. When do they next meet?

Relative speed = $6 - 4 = 2$ m/s. The faster gains a full lap (400 m) on the slower.
Time = $400/2 = 200$ seconds.

How CDS Tests This Topic

Seven archetypes: (1) basic $d = vt$, (2) average speed with two legs, (3) train crossing pole or platform, (4) two trains crossing (same or opposite direction), (5) boats and streams, (6) pursuit / head-start meeting, (7) circular track. Recognise the archetype in 5 seconds and the formula in another 5.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — Unit Conversion Cold

72 km/h = 20 m/s. 108 km/h = 30 m/s. 36 km/h = 10 m/s. 54 km/h = 15 m/s. Memorise these — they cover most CDS train problems.

Shortcut 2 — Harmonic vs Arithmetic Mean

Equal distances → harmonic. Equal times → arithmetic. Never confuse the two.

Shortcut 3 — Relative Speed Rule

Same direction: subtract. Opposite direction: add. Use for trains, pursuits, and circular tracks.

Shortcut 4 — Boat Inverse Identity

Boat from $u, d$ $$u = \frac{d + u_{up}}{2}, \quad s = \frac{d - u_{up}}{2}$$

Shortcut 5 — Ratio of Speeds, Ratio of Times

For the same distance, ratio of times = inverse ratio of speeds. If $v_1 : v_2 = 3 : 5$, then $t_1 : t_2 = 5 : 3$.

Common Question Patterns

Pattern 1 — Train Crosses Pole / Platform

Time = (length involved) / speed. For pole, length is just the train. For platform, train + platform. Convert speed to m/s first.

Pattern 2 — Two Trains Crossing

Time = (sum of lengths) / (relative speed). Same direction → subtract speeds. Opposite → add.

Pattern 3 — Average Speed Two Legs

Harmonic mean for equal distance. Arithmetic mean for equal time.

Pattern 4 — Boats and Streams

Find still-water speed and stream speed from given upstream/downstream observations. Use $u = (d + u)/2$ and $s = (d - u)/2$.

Pattern 5 — Circular Track Meeting

Same direction: meet after $L / (v_1 - v_2)$. Opposite: meet after $L / (v_1 + v_2)$. At start: LCM of lap times.

Preparation Strategy

Week 1. Master unit conversion km/h ↔ m/s cold. Drill 20 basic $d = vt$ problems. Practice average speed on equal-distance and equal-time problems.

Week 2. Trains (pole, platform, train-train) and boats. Drill the relative-speed formula until it is reflex. Cover the pursuit and circular-track archetypes.

Mock testing. Take timed CDS papers. Track unit-conversion slip-ups and direction errors (same vs opposite). Use CDS mock tests for pace.

Cross-train with Ratio and Proportion (speed ratio = inverse time ratio) and Average (harmonic mean for average speed).

Drill TSD at Speed

CDS mocks with trains, boats, pursuits, circular tracks. Seven archetypes — seven shortcuts — reflex.

Start Free Mock Test

Frequently Asked Questions

How do I convert km/h to m/s quickly?

Multiply by $\tfrac{5}{18}$. Common values: 36 km/h = 10 m/s; 54 km/h = 15 m/s; 72 km/h = 20 m/s; 90 km/h = 25 m/s; 108 km/h = 30 m/s. Memorise these for instant access in train problems.

What is the formula for two trains crossing each other?

Time = (sum of lengths) / relative speed. Relative speed is sum of speeds for opposite directions, difference for same direction. Convert all speeds to a common unit (typically m/s) first.

Why is average speed not the arithmetic mean for equal distances?

Equal distances at different speeds take different times. The slower leg takes longer, so it weighs more in the total time. The correct formula is the harmonic mean: $\bar{v} = 2 v_1 v_2 / (v_1 + v_2)$. For equal times, the arithmetic mean is correct.

How do boat-and-stream problems work?

Boat's downstream speed = (still water + stream); upstream speed = (still water − stream). From observed downstream $d$ and upstream $u$: still-water speed = $(d + u)/2$; stream speed = $(d - u)/2$. CDS uses clean integer values.

When do two runners on a circular track meet again?

Same direction: after $L / (v_1 - v_2)$, the faster has gained one full lap on the slower. Opposite direction: after $L / (v_1 + v_2)$, they have together covered one lap. Meeting at the starting point: LCM of individual lap times.

How do I solve pursuit problems?

The faster starts behind the slower or starts later. The faster closes the gap at the relative speed ($v_1 - v_2$). Time to catch up = gap / relative speed. If B starts $t_0$ hours after A: gap at start = $v_A \cdot t_0$; catch-up time = $v_A t_0 / (v_B - v_A)$.

Which CDS Maths topics connect to TSD?

Ratio and Proportion — for "same distance, ratio of speeds = inverse ratio of times". Average — for harmonic mean of speeds. Time and Work — same rate-based reasoning applied to work output.