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The distance between Delhi and Agra is $200km$. A train travels the first $100km$ at a speed of $50km/h$. How fast must the train travel the next $100km$, so as to average $70km/h$ for the whole journey?

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Last updated date: 20th Jun 2024
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Answer
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Hint: We can calculate the time to cover the first half of the journey, and the time to cover the second half can be obtained in terms of the unknown speed. The total time will be the sum of the two times. Dividing the total distance by the total time, we will get the average speed with which we can obtain an equation in terms of the unknown speed solving which we will get the final answer.

Complete step-by-step solution:
Let the next $100km$ be travelled by the train with a speed of $v{\text{ k}}m/h$
We know that the average velocity is defined as the total distance covered by an object divided by the total time taken. According to the question, the train travels between Delhi and Agra which are separated by a distance of $200km$. So the total distance covered by the train is equal to $200km$, that is we have
$d = 200km$...............(1)
Now, the journey of the train is split into two equal halves of the total distance to be covered. The first half distance of $100km$ is covered with a speed of $50km/h$. So the time taken to complete the first half of the journey is given by
${t_1} = \dfrac{{100km}}{{50km/h}}$
$ \Rightarrow {t_1} = 2h$............(2)
Also, according to our assumption, the next half distance of is covered with a speed of . So the time taken to complete the second half of the journey is given by
${t_2} = \dfrac{{100km}}{{v{\text{ }}km/h}}$
$ \Rightarrow {t_2} = \dfrac{{100}}{v}h$ ……………...(3)
So the total time of the complete journey is given by
$t = {t_1} + {t_2}$
Putting (2) and (3) in the above equation, we get
$t = 2h + \dfrac{{100}}{v}h$
$ \Rightarrow t = \left( {2 + \dfrac{{100}}{v}} \right)h$..............(4)
Now, according to the definition of the average speed, it is given by
${v_{avg}} = \dfrac{d}{t}$
Putting (1) and (4) in the above equation, we get
${v_{avg}} = \dfrac{{200km}}{{\left( {2 + \dfrac{{100}}{v}} \right)h}}$
$ \Rightarrow {v_{avg}} = \dfrac{{200}}{{\left( {2 + \dfrac{{100}}{v}} \right)}}km/h$
Now, according to the question, the average speed of the train for the whole journey is equal to $70km/h$. Therefore substituting ${v_{avg}} = 70km/h$ in the above equation, we have
$70 = \dfrac{{200}}{{\left( {2 + \dfrac{{100}}{v}} \right)}}$
$ \Rightarrow 2 + \dfrac{{100}}{v} = \dfrac{{200}}{{70}}$
Subtracting $2$ from both the sides, we have
$\dfrac{{100}}{v} = \dfrac{{200}}{{70}} - 2$
 \[ \Rightarrow \dfrac{{100}}{v} = \dfrac{6}{7}\]
Taking reciprocal of both the sides, we have
 \[\dfrac{v}{{100}} = \dfrac{7}{6}\]
Multiplying both the sides by $100$, we finally get
$v = \dfrac{{700}}{6}$
$ \Rightarrow v = 166.67km/h$

Hence, the required speed of the train to travel the next $100km$ is equal to $166.67km/h$.

Note: Do not convert the distances and the speeds given in this question to their SI units, since all of them belong to the single system of units. In this question, the motion of the train was unidirectional. That’s why we calculated the average speed, otherwise we would have calculated the average velocity for which we would have considered the total displacement of the train.