# Help understanding train problem

A train $150$ $m$ long passes a km stone in $15$ seconds and another train of the same length traveling in opposite direction in $8$ seconds. The speed of the second train is:

The solution is also provided in this link But still I could not able to understand,

Why do we need to consider the relative speed. Dose the question really says that the trains were passing at the same time?

And In solution why didn't they add a km with the trains length?

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First, the a in the phrase a km stone is just the indefinite article, not a symbol for an unspecified number. A kilometre stone is just a stone marking a particular point on a road or railway; it could just as well have been a particular telephone pole or any other fixed marker.

The first train passes the stone in $15$ seconds; it’s $150$ m long, so it’s travelling at a speed of $\frac{150}{15}=10$ m/s. Now suppose that the other train’s speed is $x$ m/s. Then the combined speed of the two trains towards each other is $x+10$ m/s. It will take them exactly as long to pass each other as it would if one were stationary and the other were going $x+10$ m/s. That’s a slightly easier problem to visualize. When you think in those terms, you can see that since both are $150$ m long, the moving train must travel $150+150=300$ m in order to pass the other train completely.

At a speed of $x+10$ m/s this takes $\frac{300}{x+10}$ seconds. We’re told that it takes $8$ seconds, so

$$\frac{300}{x+10}=8\;,$$

and therefore $8(x+10)=300$, i.e., $8x+80=300$. Solving that equation, we find that $x=\frac{55}2$ m/s, or $$\frac{55}2\cdot\frac{3600}{1000}=99\text{ km/hr}\;.$$

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:- Thanks A Lot for the answer and it is straight and to the point. –  Rasmi Ranjan Nayak Jan 31 '13 at 19:09
@Rasmi: You’re welcome. –  Brian M. Scott Jan 31 '13 at 20:49

To pass a km stone, the train needs to cover its length $=150 m$ in $15$ seconds

So, in $1$ second the 1st train travels $10m$

So, in $8$ second the 1st train travels $10m\cdot 8=80m$

To pass another train travelling in opposite direction, both train together need to cover the sum of their length $=2\cdot150m=300m$ in $8$ seconds

So, the other train needs to travel the rest length i.e., $(300-80)m=220 m$ in $8$ seconds

So, in $1$ second the 2nd train travels $\frac{220m}8=27.5m$

So, the speed of the 2nd train is $22.5m/s$ or $22.5\cdot\frac{3600}{1000}km/h=99km/h$

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Thanks for a explanation. But I would like to know why did n't he add 1 km length to the train's length? –  Rasmi Ranjan Nayak Jan 31 '13 at 18:31
@RasmiRanjanNayak, the length of km stone is very very lees than $1$ km, also negligible as compared to the length of the train. –  lab bhattacharjee Jan 31 '13 at 18:33
:- I think, Brian M. Scott's solution below gives the glimpse the answer of my chaos. –  Rasmi Ranjan Nayak Jan 31 '13 at 19:11