Important Facts and Formulas: Problems Based on Trains | General Test Preparation for CUET - CUET Commerce PDF Download

Introduction

Issues concerning trains are akin to challenges associated with 'Speed, Time, and Distance,' and certain principles of 'Speed, Time, and Distance' are applicable to both. The sole distinction lies in the inclusion of the length of the moving object (train) in these particular scenarios.

Important Facts and Formulae

• km/hr to m/s conversion:
  
• m/s to km/hr conversion:
  

Formulas for finding Speed, Time and Distance

• Time taken by a train of length l metres to pass a pole or standing man or a signal post is equal to the time taken by the train to cover l metres.
• Time taken by a train of length l metres to pass a stationery object of length b metres is the time taken by the train to cover (l + b) metres.
• Suppose two trains or two objects bodies are moving in the same direction at u m/s and v m/s, where u > v, then their relative speed is = (u - v) m/s.
• Suppose two trains or two objects bodies are moving in opposite directions at u m/s and v m/s, then their relative speed is = (u + v) m/s.
• If two trains of length a metres and b metres are moving in opposite directions at u m/s and v m/s, then:
  The time taken by the trains to cross each other :
  
• If two trains of length a metres and b metres are moving in the same direction at u m/s and v m/s, then:
  The time taken by the faster train to cross the slower train:
  
• If two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then:
  (A's speed) : (B's speed) =(√b : √a)

Rules to Solve

Rule 1

The speed of the train is calculated by using following formula:
 

Example: Let's say a train covers a distance of 150 kilometers in 3 hours. We can use the formula to find the speed of the train.

Using the formula Speed of Train (S)=Distance Covered (d) /Time Taken (t)

with the distance covered (d) being 150 kilometers and the time taken (t) being 3 hours, the speed of the train (S) is calculated to be 50 kilometers per hour that is 150/3 

Rule 2

• Relative Speed of a stationary object and moving object = Speed of the moving object.
• Relative Speed of two objects moving in opposite direction = Sum of their speeds.
• Relative Speed of two objects moving in same direction = Difference of their speeds.

Rule 3

• Distance covered by train in passing a pole or a stationary man or a signal post or any other object of negligible length = Length of the train.
• Distance covered by train in passing a stationary object (bridge, platform etc) having same length = Sum of lengths of the train and that particular stationary object.

Rule  4

• Two trains of length and y are moving in opposite direction with speed u and v respectively, then time taken by trains to cross each other
• Two trains of length and y are moving in same direction with speed u and v respectively, then time taken by the faster train to cross the slower train=

TIPS to Crack the Questions

Tip 1

Understand the concepts involved in a train crossing a pole or a platform

1. Time taken by a train of length L to pass a pole or standing man or a signal post is equal to the time taken by the train to cover distance L.

2. Time taken by a train of length L to pass a station of length b is the time taken by the train to cover the distance (L + b).

Example 1: A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?
Sol:
Time taken to cross a pole = time taken to cover distance equal to its own length.
Speed of the train in m/s = 60 x 5 / 18 = (50/3) m/s.
Length of the train = Speed of the train x Time taken to cross the pole
= 50/3 x 9
=   Length of the train = 150m.

Example 2: A train passes a station platform in 36 sec and a man standing on the platform in 20 sec. If the speed of the train is 54 km/hr, what is the length of the platform?
Sol:
Speed of the train in m/s = 54 x 5 / 18 = 15m/s.
Length of the train = 15 x 20 = 300m.
Distance traveled in 36s = 15 x 36 = 540m. (This is the length of the train + platform combined)
Length of the platform = (540 – 200) m = 240m.

Tip 2

For problems on 2 trains, use the concept of relative velocity

1. If two trains of length a and b are moving in opposite directions at u m/s and v m/s, then time taken by the trains to cross each other = (a + b) / (u + v).

2. If two trains of length a and b are moving in the same direction at u m/s and v m/s, then time taken by the faster train to cross the slower train = (a + b) / (u - v) .

Example 1: 2 trains of length 137 m and 163 m are running towards each other and speeds 42 km/hr and 48 km/hr respectively. In what time will the two trains cross each other?
Sol:
Relative velocity = 42 + 48 = 90 km/hr = (90 x 5/18) m/s = 25 m/s. (Opposite directions)
Time taken to cross each other = 300 / 25 = 12 sec.

Example 2: 2 trains running in opposite directions cross a man standing on the platform in 27s and 17s respectively and they cross each other in 23s. Find the ratio of their speeds.
Sol:
Let the speeds be x m/s and y m/s respectively.
Then, length of 1st train = 27x and that of 2nd train = 17y.
Time taken to cross each other = (27x + 17y) / (x + y) = 23.
=  27x + 17y = 23x + 23y.
= 4x = 6y.
=  x: y = 3 : 2.

Tip 3

Calculation of time of rest per hour 

Example: Without stoppage, the speed of the train is 54km/hr and with stoppage, it is 45km/hr. For how many minutes, does the train stop per hour?

Sol:
 

Tip 4

Calculation of time taken by trains to cross each other, if two trains having equal length and different speeds take time t1 & t2  to cross a pole

I. If trains are moving in opposite direction:

II. If trains are moving in same direction:
 

Example: -Two trains of equal lengths take 4s & 5s respectively to cross a pole. If these trains are moving in the same direction then how long will they to cross each other?

Sol:

Miscellaneous Questions 

1. A train 300 m long is running at a speed of 54 km/hr. In what time will it pass a bridge 150 m long?

Solution: 

Speed = {54 × (5/18)} m/sec = 15 m/sec

Total distance which needs to be covered = (300+150)m = 450m

Time = Distance/Speed

Required Time = 450 / 15 = 30 seconds

2. Time is taken by two trains running in opposite directions to cross a man standing on the platform in 28 seconds and 18 seconds respectively. It took 26 seconds for the trains to cross each other. What is the ratio of their speeds?

Solution: 

Let the speed one train be x and the speed of the second train be y

Length of the first train = Speed × Time = 28x

Length of second train =  Speed × Time = 18y

So, {(28x+18y) / (x+y)} = 26

⇒ 28x+18y = 26x+26y

⇒ 2x = 8y

Therefore, x:y = 4:1

3.  A train running at the speed of 56 km/hr crosses a pole in 18 seconds. What is the length of the train?

Solution:

Speed = {56 × (5/18)} m/sec = (140/9) m/sec 

Length of the train (Distance) = Speed × Time = {(140/9) × 18} = 280 m