Question

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?

Answer 1

Hint: You can start by briefly explaining distance and speed. Then write the equation for speed, i.e. \[s = \dfrac{d}{t}\]. Then apply this equation for the first half of travel, the second half of the travel and then for the overall travel and obtain the time taken in each case. We know that the time taken for the overall travel will be equal to the sum of the time taken for the first and second half of the travel. Use this relation to reach the solution.

Complete step by step answer:
Distance – It is a scalar quantity which measures the amount or extent of space between two objects or points.
Speed – Speed is the distance that an object covers in unit time. The SI unit of speed is \[km/h\] . It is also a scalar quantity.
We know the speed of any object can be calculated by
\[s = \dfrac{d}{t}\]
Here, \[s = \] Speed of the body
\[d = \] Distance covered
\[t = \] Time taken to cover that distance
Let the time taken to cover the first \[100km\] , the second \[100km\] and the total time taken be \[{t_1}\] , \[{t_2}\] and \[{t_t}\] respectively.
So, \[{t_t} = {t_1} + {t_2}\]
Also let the speed of the train while covering the second \[100km\] be \[S\] .
For the first \[100km\] equation of speed becomes
\[50 = \dfrac{100}{{t_1}}\]
\[ \Rightarrow {t_1} = 2h\]
For the second \[100km\] equation of speed becomes
\[S = \dfrac{{100}}{{{t_2}}}\]
\[{t_2} = \dfrac{{100}}{S}h\]
For the overall motion of the train the equation of speed becomes
\[70 = \dfrac{{200}}{{{t_t}}}\]
\[{t_t} = \dfrac{{200}}{{70}}h\]
Now, we know \[{t_t} = {t_1} + {t_2}\]
\[ \Rightarrow \dfrac{{200}}{{70}} = 2 + \dfrac{{100}}{S}\]
\[ \Rightarrow \dfrac{{200}}{{70}} - 2 = \dfrac{{100}}{S}\]
\[ \Rightarrow \dfrac{{60}}{{70}} = \dfrac{{100}}{S}\]
\[ \Rightarrow S = 116.66km/h\]
Hence, the train needs to travel with a speed of \[116.66km/h\] during the second half of the travel to make the average speed of the train \[70km/h\] .

Note:
In this particular problem we considered speed and distance and used the equation \[S = \dfrac{d}{t}\] to get the result, if the problem went something like this, a train travels the first \[100km\] at a speed of \[50km/h\] towards north, then we would have used the equations for velocity, i.e.
 Velocity \[ = \dfrac{{Displacement}}{{Time}}\] . It is crucial to remember this differentiation.