SSC CGL Maths · Time, Speed and Distance

Trains and Relative Speed

A train has length, so to pass anything it must cover its own length plus the length of what it passes, at the relative speed of the two.

All Time, Speed and Distance notes

Why this matters

Eight questions in the bank, seven of them EASY, and none from the 2025 shifts. Each one is two numbers to get right: the distance, meaning which lengths to add, and the speed, meaning the train's own, a difference or a sum. The m/s values of multiples of 18 km/h do most of the arithmetic.

Concept 1 of 2: A train passing a pole, a platform or a standing train

A train has passed a pole only when its last coach goes by, so its engine has moved one full train length. To clear a platform the engine must run the platform's length and then the train's own length again.

Definition

A pole, a signal or a person standing still: distance $=$ the train's length.
A platform, bridge, tunnel or a train standing on the next line: distance $=$ the train's length $+$ the other length.
Time $= \frac{\text{distance}}{\text{speed}}$ , with lengths in metres and the speed in m/s.
Two crossings give the speed: a train that passes a pole in $t_1$ and a platform of length $p$ in $t_2$ has speed $\frac{p}{t_2 - t_1}$ , because the extra time is spent on the platform alone.
Fast route: a speed that is a multiple of 18 km/h is a whole number in m/s: 54 km/h is 15 m/s, 108 km/h is 30 m/s.

Train passing a fixed object

t = \frac{L_{\text{train}} + L_{\text{object}}}{s} \qquad s = \frac{p}{t_2 - t_1}

$L_{\text{object}}$ length of the platform, bridge or standing train (0 for a pole)
$s$ speed of the train in m/s
$p$ length of the platform

Worked example

A 240 m train runs at 72 km/h. How long does it take to cross a 400 m bridge?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 14 Jul 2023, Shift 1 · Q60Easy

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

A train 900 m long is running at 108 km/h. How long will it take to clear a 900 m long platform completely?

Leaving out the train's own length

To clear a platform the train covers the platform and its own length. Using the platform alone gives a time that is too short.

Adding a length for a pole or a standing man

A pole or a person has no length worth counting, so the distance is the train's length alone.

Drill 1 more on a train passing a pole, a platform or a standing train

Concept 2 of 2: Trains passing moving trains or people

If the thing being passed is moving too, use relative speed exactly as in a chase: the difference when both go the same way, the sum when they come towards each other. The distance is still the lengths added.

Definition

Two trains, same direction: time $= \frac{L_1 + L_2}{s_1 - s_2}$ .
Two trains, opposite directions: time $= \frac{L_1 + L_2}{s_1 + s_2}$ .
A train passing a moving person: the person adds no length, so the time is the train's length over the relative speed: the difference if the person walks the same way, the sum if towards the train.
Both crossing times given: the lengths over the same-direction time give $s_1 - s_2$ , and over the opposite-direction time give $s_1 + s_2$ ; half the sum and half the difference give the speeds.
Add or subtract the speeds in km/h first and change the result to m/s once.

Two trains crossing

t_{\text{same}} = \frac{L_1 + L_2}{s_1 - s_2} \qquad t_{\text{opp}} = \frac{L_1 + L_2}{s_1 + s_2}

$L_1, L_2$ lengths of the two trains
$s_1, s_2$ speeds of the two trains, the first one the faster

Worked example

A 200 m train at 75 km/h overtakes a 175 m train running at 48 km/h on a parallel track. How long does the overtaking take?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 19 Sep 2024, Shift 1 · Q71Moderate

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

Two trains of lengths 150 m and 250 m run on parallel lines. When they run in the same direction, they take 20 seconds to cross each other and, when they run in the opposite direction, they take 5 seconds to cross each other. What is the speed of the two trains?

Using one train's length for two moving trains

Two trains have crossed only when their last coaches part, so the distance is the sum of both lengths.

Subtracting the speed of a walker coming towards the train

A walker coming towards the train meets it sooner, so the speeds add. Subtract only when the walker goes the same way as the train.

Drill 5 more on trains passing moving trains or people

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)

A train passing a pole, a platform or a standing train
Train passing a fixed object
$t = \frac{L_{\text{train}} + L_{\text{object}}}{s} \qquad s = \frac{p}{t_2 - t_1}$
Trains passing moving trains or people
Two trains crossing
$t_{\text{same}} = \frac{L_1 + L_2}{s_1 - s_2} \qquad t_{\text{opp}} = \frac{L_1 + L_2}{s_1 + s_2}$

Watch out for (4)

Leaving out the train's own length
→ A train passing a pole, a platform or a standing train
Adding a length for a pole or a standing man
→ A train passing a pole, a platform or a standing train
Using one train's length for two moving trains
→ Trains passing moving trains or people
Subtracting the speed of a walker coming towards the train
→ Trains passing moving trains or people

Test yourself on Time, Speed and Distance

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Time, Speed and Distance chapter test Drill all 8 questions