## Introduction to Two Trains Moving in Opposite Directions

In this section, we will learn the concept of two trains heading in opposite directions. The concept behind it is relative speed. We can define relative speed as the speed of a moving body concerning another. The train on the left is going to the north, and the one on the right is going to the south. First, let's start with what happens when both trains are at rest. When both trains are at rest, they are stationary, and their speeds equal zero. Next, let's see what happens when two trains are moving at equal speeds in opposite directions.

Two Trains Coming From the Opposite Direction

## What is Relative Speed?

We can define relative speed as the speed of a moving body with respect to another. When two bodies move in the same direction, their relative speed is computed by their difference. But when two bodies move in the opposite direction, the relative speed is calculated by adding the speed of both bodies.

## Two Trains Move in Opposite Directions Along Parallel Tracks

Two Trains concept is a mathematical analogy that describes the motion of two trains running in opposite directions on parallel tracks; these trains will never collide. It also shows that if one train changes its direction, it collides with the other. The "Two Trains" concept is a popular analogy for parallel lines in mathematics. It features two trains travelling down parallel tracks in opposition to one another. The trains pass one another and proceed on their routes after they come together.

In this article, we shall study the idea of two trains passing one another:

When two train passes a moving object (having some length) in the opposite direction

Let the length of the faster train be l meters, and the length of the slower train be m meters

Let the speed of a faster train be x km/hr

Relative speed = (x + y) km/hr.

Then, the time is taken by the faster train to pass the slower train $= \dfrac{(l + m)meter}{(x + y) km/hr}$

Now we will learn to calculate when two trains run on parallel tracks (having some length) in the opposite direction.

Two trains, A and B, come from opposite directions; Train A crosses B in 2 minutes, and the length of train $A$ is double the length of train B. The speeds of train A and B are $50 \mathrm{~km} / \mathrm{hr}$ and $22 \mathrm{~km} / \mathrm{hr}$ respectively.

Ans:

Formula used:

Time taken by train $\mathrm{A}$ of length ' $a$ ' to cross train $\mathrm{B}$ of length ' $b$ ' running in opposite direction $=\dfrac{(a+ b)}{\left(v_1+v_2\right)}$ where $v_1$ and $v_2$ are the speeds of trains $A$ and $B$ respectively.

Distance $=$ speed $\times$ time

Let the length of Train A be '2l' $m$, and the Length of Train B be 'l' $m$.

$\Rightarrow \text { Relative speed }=(50+22) \mathrm{kmph}=72 \mathrm{kmph}=20 \mathrm{~m} / \mathrm{s}$

$\Rightarrow \text { Time }=2 \text { minutes }=120 \text { seconds. }$

$\Rightarrow(\mathrm{l}+2 \mathrm{l})=20 \times 120$

$\Rightarrow 3 \mathrm{l}=2400$

$\Rightarrow \mathrm{l}=800 \mathrm{~m}$

Thus, length of Train $\mathrm{A}=2 \mathrm{l}=1600 \mathrm{~m}$

## Two Trains are Heading in Opposite Directions Related Solved Examples

Here are some examples related to two trains running in opposite directions, which are like this;

Q 1. Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds, respectively and they cross each other in 23 seconds. The ratio of their speed is?

Ans: According to the problem statement, the time of crossing of a man standing on the platform is 27 seconds and 17 seconds. Let the speed of one train be $x$, and the speed of another train be $y$ in meters per second

As speed is defined as the ratio of distance and time. So, distance is the product of speed and time. This can be mathematically expressed as:

$\text{speed}=\dfrac{\text { distance }}{\text { time }}$

$\text { distance }=\text { speed } \times \text { time }$

Therefore, distance travelled by first train $=27 x \mathrm{m}$.

Distance travelled by second train $=17 \mathrm{y} \mathrm{m}$.

Also, the time of crossing over by two trains is 23 seconds. Now, time can be related as

$\text { time }=\dfrac{\text { distance }}{\text { speed }}$

The total distance travelled $(27 x+17 y) m$.

The total speed is the relative speed, and since both trains move in opposite directions, their speed is added. So, the final speed for crossover $=x+y$

Therefore, the time is taken to cross over

$\dfrac{27 x+17 y}{x+y}=23$

$27 x+17 y=23 x+23 y$

$27 x-23 x=23 y-17 y$

$4 x=6 y$

$\dfrac{x}{y}=\dfrac{6}{4}=\dfrac{3}{2}$

Hence, the obtained ratio is $\dfrac{3}{2}$.

Q 2. With speeds of 34 km/hr and 38 km/hr, respectively, two 250 m and 150 m long trains travel on parallel lines in opposing directions. After what duration will they cross each other?

Solution- Relative speed of train = (34 + 38) km/hr

= 72 km/hr

$= 72 \times \dfrac{5}{18}$ m/sec

= 20 m/sec

Time required for two trains to cross each other = Sum of train lengths upon the relative speed of trains

$= \dfrac{(250+150)}{20}$

$= \dfrac{400}{20}$

= 20 Sec.

Consequently, the 20-second distance between the two trains was crossed.

Thus the trains cross each other at 20 sec.

Q 3. With speeds of 54 km/hr and 36 km/hr, two 162 m and 158 m long trains travel on parallel lines in opposing directions. After what duration will they cross each other?

Solution- Relative speed of train = (54 +36) km/hr

= 90 km/hr

$= 90 \times \dfrac{5}{18}$ m/sec

= 25 m/sec

Time required for two trains to cross each other = Sum of train lengths upon the relative speed of trains

$= \dfrac{(162+158)}{25}$

$= \dfrac{320}{20}$

= 16 Sec.

Consequently, the 16-second distance between the two trains was crossed.

Thus the trains cross each other at 16 sec.

## Two Trains are Moving in Opposite Directions Related Practice Problems

Here are some practice problems related to two trains running in opposite directions, which are like this;

Q 1. With speeds of 56km/hr and 52 km/hr, respectively, two trains that are 136 m and 242m long travel on parallel lines in opposing directions. After what duration will they cross each other?

Ans. 12.6 sec.

Q 2. With speeds of 50km/hr and 40 km/hr, respectively, two trains that are 126 m and 158 m long travel on parallel lines in opposing directions. After what duration will they cross each other?

Ans. 11.36 sec.

Q 3. Two trains are moving at equal speeds in the opposite direction along two parallel railway tracks. If the wind is blowing with speed $u$ along the track so that the relative velocities of the trains for the wind are in the ratio $1: 2$, then the speed of each train must be?

Ans. 3u

## Summary

This article mainly learned about two trains running in opposite directions, one of the important topics for Mathematics. This article discusses how two trains are moving with equal speeds in opposite directions and two trains, a and b coming from the opposite direction. Due to this reason, examples and practice problems related to two trains heading in opposite directions have been given in this article, but their concept has also been talked about. We wish this article will prove to be very useful from the point of view of examination and knowledge.

## FAQs on Two Trains are Moving in Opposite Directions

1. What happens when two trains are travelling in the same direction?

Two trains' speeds are subtracted when travelling in the same direction. Two trains' speeds are added when they are travelling in opposite directions. The total distance in the two situations mentioned above is equal to the sum of the lengths of the two trains.

2. What is the speed formula?

The speed of an object is calculated as the total distance travelled divided by the total travel time. The total distance travelled and travel time is denoted by d and t respectively, i.e. $\dfrac{d}{t}$. SI unit of speed is represented by meters per second.

3. What transpires when two trains cross paths?

A sound like slapping will be made between the trains as they pass if they are going quickly enough and are in the proper form. The ensuing slapping sound might sound like a sonic boom if they both travel at roughly half the speed of sound.