Question

Two trains one travelling 15 kmph faster than other, leave the same station at the same time, one travelling east and other travelling west. At the end of 6 hours they are 570km apart. What are the speeds of each train?

Answer 1

Hint: Let us take a rough figure that represents the given information as follows.

We solve this problem by assuming the speeds of trains as some variables. Then we find the distance traveled by both the trains after 6 hours and add then to get total of 570km.
We use the formula of speed is given as
\[\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}\]
Then we get two equations in two variables so that we can solve for speeds easily.
Complete step by step answer:
Let us assume that the speeds of train 1 and train 2 as \[{{V}_{1}},{{V}_{2}}\] respectively.
We are given that one travelling 15 kmph faster than other
Let us assume that train 2 is faster than train 1 then we get
\[\Rightarrow {{V}_{2}}={{V}_{1}}+15........equation(i)\]
We are given that at the end of 6 hours they are 570km apart
Let us assume that the distance between two trains as
\[\Rightarrow AB=570km\]
Let us assume that the distance travelled by train 1 in 6 hours as OB

We know that the formula of speed is given as
\[\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}\]
By using the above formula we get the distance travelled by train 1 as
\[\begin{align}
  & \Rightarrow {{V}_{1}}=\dfrac{OB}{6} \\
 & \Rightarrow OB=6{{V}_{1}} \\
\end{align}\]
Let us assume that the distance travelled by train 1 in 6 hours as OA
\[\begin{align}
  & \Rightarrow {{V}_{2}}=\dfrac{OA}{6} \\
 & \Rightarrow OA=6{{V}_{2}} \\
\end{align}\]
From the figure we can say that the distance AB is divided into two parts as that is
\[\Rightarrow AB=OA+OB\]
Now, by substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow 570=6{{V}_{2}}+6{{V}_{1}} \\
 & \Rightarrow {{V}_{1}}+{{V}_{2}}=95 \\
\end{align}\]
Now, let us substituting the value of \[{{V}_{2}}\] from equation (i) in above equation we get
\[\begin{align}
  & \Rightarrow {{V}_{1}}+{{V}_{1}}+15=95 \\
 & \Rightarrow 2{{V}_{1}}=80 \\
 & \Rightarrow {{V}_{1}}=40 \\
\end{align}\]
By substituting this value in equation (i) we get
\[\begin{align}
  & \Rightarrow {{V}_{2}}=40+15 \\
 & \Rightarrow {{V}_{2}}=55 \\
\end{align}\]
Therefore we can conclude that the speeds of two trains are 40kmph and 55kmph.

Note:
 We can solve this problem in another method.
Let us assume that the speeds of train 1 and train 2 as \[{{V}_{1}},{{V}_{2}}\] respectively.
We are given that one traveling 15 kmph faster than other
Let us assume that train 2 is faster than train 1 then we get
\[\Rightarrow {{V}_{2}}={{V}_{1}}+15........equation(i)\]
We are given that at the end of 6 hours they are 570km apart
We use the relative velocity concept.
The relative speed of two trains when they are moving in opposite directions is given as
\[\Rightarrow {{V}_{rel}}={{V}_{1}}+{{V}_{2}}\]
Now, let us substituting the value of \[{{V}_{2}}\] from equation (i) in above equation we get
\[\begin{align}
  & \Rightarrow {{V}_{rel}}={{V}_{1}}+{{V}_{1}}+15 \\
 & \Rightarrow {{V}_{rel}}=2{{V}_{1}}+15 \\
\end{align}\]
Here, the relative speed is obtained by using the total distance between two trains and total time that is
\[\begin{align}
  & \Rightarrow {{V}_{rel}}=\dfrac{570}{6} \\
 & \Rightarrow {{V}_{rel}}=95 \\
\end{align}\]
By substituting the value of \[{{V}_{rel}}\] in above equation we get
\[\begin{align}
  & \Rightarrow 2{{V}_{1}}+15=95 \\
 & \Rightarrow {{V}_{1}}=\dfrac{80}{2} \\
 & \Rightarrow {{V}_{1}}=40 \\
\end{align}\]
By substituting this value in equation (i) we get
\[\begin{align}
  & \Rightarrow {{V}_{2}}=40+15 \\
 & \Rightarrow {{V}_{2}}=55 \\
\end{align}\]
Therefore we can conclude that the speeds of two trains are 40kmph and 55kmph.