Episode 8 — Aptitude and Reasoning / 8.12 — Problems on Trains
8.12.a Concepts and Formulas -- Problems on Trains
1. Why Trains are Special
Unlike cars or people, trains have a significant physical LENGTH. When we say a train "crosses" something, the ENTIRE train must pass by. This means:
The distance covered = Length of the train + Length of the object
(If the object has no length, like a pole, only the train's length matters.)
2. Train Crossing a Pole (or a Standing Person)
A pole or a person standing on the platform has negligible length compared to a train. So the distance the train covers to "cross" the pole is simply its own length.
Scenario:
+-----------+
| TRAIN |------> speed S
+-----------+
|
POLE
The train starts crossing when its front reaches the pole.
The train finishes crossing when its rear passes the pole.
Distance covered = Length of train (L)
Formula
+---------------------------------------+
| |
| Time = Length of train / Speed |
| |
| T = L / S |
| |
+---------------------------------------+
Diagram: Before, During, After
BEFORE:
+-----------+ |
| TRAIN |----> | POLE
+-----------+ |
CROSSING (front at pole):
+-----------+
| TRAIN |---->
+-----------+
|
POLE
CROSSED (rear past pole):
+-----------+
| | TRAIN |---->
| +-----------+
POLE
Distance = Full length of train from "front at pole" to "rear past pole"
3. Train Crossing a Platform (or Bridge)
A platform or bridge has its own length. The train must cross both its own length AND the platform's length.
+-----------+
| TRAIN |------> speed S
+-----------+
|<------- Platform ------->|
===========================
Formula
+---------------------------------------------------+
| |
| Time = (Length of train + Length of platform) / S |
| |
| T = (L_train + L_platform) / S |
| |
+---------------------------------------------------+
Diagram: Start to Finish
START (front of train reaches start of platform):
+-----------+
| TRAIN |===========================
+-----------+ PLATFORM
|<------- L_p ------------>|
FINISH (rear of train clears end of platform):
+-----------+
=========================== | TRAIN |
PLATFORM +-----------+
Total distance = L_train + L_platform
4. Train Crossing Another Train -- Opposite Direction
When two trains move towards each other, their relative speed is the SUM of their individual speeds.
Train A Train B
+--------+ +--------+
| L_a |-----> <-------| L_b |
+--------+ S_a S_b +--------+
Relative speed = S_a + S_b
Formula
+---------------------------------------------------+
| |
| Time = (L_a + L_b) / (S_a + S_b) |
| |
+---------------------------------------------------+
Why L_a + L_b?
BEFORE CROSSING:
+--------+ +--------+
| A |-----> <-------| B |
+--------+ +--------+
|<------------ gap ----------------->|
AFTER CROSSING (completely passed each other):
+--------+ +--------+
<-------| B | | A |----->
+--------+ +--------+
The relative distance covered = L_a + L_b
(From front of A meeting front of B, to rear of A clearing rear of B)
5. Train Crossing Another Train -- Same Direction
When two trains move in the same direction, the relative speed is the DIFFERENCE of their speeds. The faster train overtakes the slower one.
Faster Train A Slower Train B
+--------+ +--------+
| L_a |-----> | L_b |----->
+--------+ S_a +--------+ S_b
(S_a > S_b)
Relative speed = S_a - S_b
Formula
+---------------------------------------------------+
| |
| Time = (L_a + L_b) / (S_a - S_b) |
| |
+---------------------------------------------------+
Diagram
BEFORE (A approaches B from behind):
+--------+ gap +--------+
| A |-----> | B |----->
+--------+ +--------+
AFTER (A has completely passed B):
+--------+ gap +--------+
| B |-----> | A |----->
+--------+ +--------+
Total relative distance = L_a + L_b
6. Man on Platform vs Man on Moving Train
This is a crucial distinction that examiners love to test.
Man Standing on Platform (Stationary Observer)
The man is stationary. The train crosses him just like crossing a pole.
Time = L_train / S_train
Man Walking on Platform
Case 1: Man walks in the SAME direction as the train
+-----------+
| TRAIN |-----> S_train
+-----------+
o----> S_man (same direction)
Relative speed = S_train - S_man
Time = L_train / (S_train - S_man)
Case 2: Man walks in the OPPOSITE direction to the train
+-----------+
| TRAIN |-----> S_train
+-----------+
<----o S_man (opposite direction)
Relative speed = S_train + S_man
Time = L_train / (S_train + S_man)
Man Sitting in Another Train
The man is part of the other train. This reduces to the
"train crossing another train" problem.
Same direction: T = (L_1 + L_2) / (S_1 - S_2)
Opposite direction: T = (L_1 + L_2) / (S_1 + S_2)
Man Sitting in the SAME Train (Looking at a Platform)
From the perspective of the man on the train, the platform approaches
at the train's speed. The man observes the platform passing by.
Time to cross platform = L_platform / S_train
NOTE: The train's own length does NOT matter here because the man is
a point observer. The platform passes by him over its own length.
This is a common trap! If the question asks how long a man on a train sees a platform, the answer uses only the platform length, not the train length.
7. Time for a Train to Cross a Man on Another Train
When the question says "a train crosses a man sitting in another train,"
the man is treated as a point. Only the LENGTH of the crossing train matters.
Same direction: T = L_crossing_train / |S1 - S2|
Opposite direction: T = L_crossing_train / (S1 + S2)
This differs from two trains crossing each other (where both lengths are added).
8. Two Trains Starting Simultaneously from Different Stations
Station A Station B
+--------+ +--------+
| T_1 |-----> D km <--------| T_2 |
+--------+ S_1 S_2 +--------+
Time to meet = D / (S_1 + S_2)
At the point of meeting:
Distance from A = S_1 x D / (S_1 + S_2)
Distance from B = S_2 x D / (S_1 + S_2)
9. Train Passing Through a Tunnel
Same as crossing a platform -- the train must cover its own length plus the tunnel length.
+-------+
| |==================| |
| Entry | <--- Tunnel ---> | Exit |
| |==================| |
+-------+ +-------+
Time = (L_train + L_tunnel) / S_train
10. Relative Speed Summary for Trains
+----------------------------------------------------------+
| Situation | Relative Speed | Distance |
|--------------------------|----------------|---------------|
| Train crosses pole | S | L |
| Train crosses platform | S | L + L_p |
| Train crosses man(same) | S - S_man | L |
| Train crosses man(opp) | S + S_man | L |
| Two trains (same dir) | S1 - S2 | L1 + L2 |
| Two trains (opp dir) | S1 + S2 | L1 + L2 |
| Train crosses man in | | |
| another train(same) | S1 - S2 | L1 only |
| Train crosses man in | | |
| another train(opp) | S1 + S2 | L1 only |
+----------------------------------------------------------+
11. Finding Length or Speed from Two Conditions
Many problems give two crossing scenarios and ask you to find the train's length or speed. Set up two equations.
Example Setup: Train crosses pole and platform
Crossing pole: L / S = T1
Crossing platform: (L + P) / S = T2
From equation 1: L = S x T1
Substitute into 2: (S.T1 + P) / S = T2
T1 + P/S = T2
P/S = T2 - T1
S = P / (T2 - T1)
Then: L = S x T1
Example Setup: Train crosses two platforms of different lengths
(L + P1) / S = T1
(L + P2) / S = T2
Subtracting: (P2 - P1) / S = T2 - T1
S = (P2 - P1) / (T2 - T1)
Then: L = S x T1 - P1
12. Speed in m/s vs km/h -- Critical for Trains
Train problems almost always mix units:
- Lengths are given in metres
- Speeds are given in km/h
- Time is asked in seconds
ALWAYS convert speed to m/s FIRST when lengths are in metres.
km/h to m/s: Multiply by 5/18
m/s to km/h: Multiply by 18/5
13. Master Formula Table
+----------------------------------------------------------+
| |
| T = (Sum of relevant lengths) / (Relative speed) |
| |
| Relevant lengths: |
| - Crossing pole/person: L_train only |
| - Crossing platform: L_train + L_platform |
| - Crossing another train: L_train1 + L_train2 |
| - Crossing man on train: L_crossing_train only |
| |
| Relative speed: |
| - Object stationary: S_train |
| - Same direction: |S1 - S2| |
| - Opposite direction: S1 + S2 |
| |
+----------------------------------------------------------+