CDS Mathematics · Time, Speed and Distance

Trains: Length and Crossing

A train passing an object must cover its own length plus the object's length, at the speed relative to that object.

All Time, Speed and Distance notes CDS Mathematics strategy

Why this matters

Fourteen PYQs. The distance to cover is always 'train + whatever it passes' (zero for a pole or a person), and the speed is always the relative speed. Most mistakes are unit conversions: multiply km/hr by 5/18 for m/s.

Concept 1 of 2: Passing poles, platforms and bridges

The front of the train must travel until the back has cleared the object. That is the train's length plus the object's length — or just the train's length for a point like a pole or a km-stone.

Definition

Pole, person, km-stone: distance $= L$ .
Platform, bridge, station of length $P$ : distance $= L + P$ .
$1$ km/hr $= \dfrac{5}{18}$ m/s; $1$ m/s $= \dfrac{18}{5}$ km/hr.
Two crossings at the same speed: subtract them to eliminate the unknown length.

Crossing time

t = \dfrac{L + P}{v}

Worked example

150

m train crosses a pole in

9

seconds. How long does it take to cross a

250

m platform?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q28Easy

Example 1 · Time, Speed and Distance · Trains and Relative Speed

A train 100 m long passes a platform 100 m long in 10 seconds. The speed of the train is

Clearing the last stone

Passing

91

km-stones spans

90

km between the first and last, plus the train's length to clear the last one completely.

Drill 6 more on passing poles, platforms and bridges

Concept 2 of 2: Trains passing moving objects

When the object also moves, use the speed of the train relative to it: the difference if they move the same way, the sum if they approach. A man sitting in another train is still a point.

Definition

Same direction: relative speed $= u - v$ . Opposite: $u + v$ .
Train passing a train: distance $= L_1 + L_2$ . Passing a man in the other train: distance $= L_1$ only.
Same-way time is $k$ times opposite-way time: $u + v = k(u - v)$ .
Overtaking two walkers of different speeds gives two expressions for the same length.

Relative crossing

t = \dfrac{L_1 + L_2}{u \pm v}

Worked example

180

m train at

60

km/hr overtakes a man walking at

6

km/hr the same way. How long does it take?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q51Moderate

Example 2 · Time, Speed and Distance · Trains and Relative Speed

Two trains

X

and

Y

are travelling in the same direction at 100 km/hr and 60 km/hr respectively. Train

X

crosses a man in train

Y

in 9 seconds. What is the length of train

X

A man in a train is a point

When a train crosses a PERSON in another train, the other train's length is a distractor. Adding it gives a speed that matches no option.

Drill 6 more on trains passing moving objects

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Passing poles, platforms and bridges
Crossing time
$t = \dfrac{L + P}{v}$
Trains passing moving objects
Relative crossing
$t = \dfrac{L_1 + L_2}{u \pm v}$

Watch out for (2)

Clearing the last stone
→ Passing poles, platforms and bridges
A man in a train is a point
→ Trains passing moving objects

Test yourself on Time, Speed and Distance

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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