Question

Point A and B are 70km apart on a highway. A car starts from A and another car starts from B at the same time. If they travel in the same direction they meet in 7 hours, but if they travel towards each other they meet in one hour. What are their speeds in km/hr?
a)40, 20
b)50, 30
c)40, 30
d)None of these

Answer 1

Hint: We need to use the principle of relative velocity in this case. Thus, when the cars travel towards each other, their effective relative velocity gets added up whereas when they travel in the same direction, their relative velocity gets subtracted. Then we can use the distance covered and speed distance formula to obtain the solution to this question.

Complete step-by-step answer:
In this case, there are two situations, one in which the cars travel towards each other and the other when the cars move away from each other. The principle of relative velocity states that
When two objects move towards each other, the relative velocity between the objects is the sum of the velocities of the two objects. …………………………….(1.1)
When two objects move in the same direction, the relative velocity between the objects is the difference of the velocities of the two objects. ……………………………(1.2)
We also use the fact that,
\[\text{Time taken for two objects to meet with each other}=\dfrac{\text{Distance between them}}{\text{Relative velocity between the objects}}................(1.3)\]
Let the velocities of the cars starting from point A and B be ${{v}_{a}}$ and ${{v}_{b}}$ respectively. The initial distance between them is given to be 70km
Now, from the question, time taken for them to meet if they travel in the same direction=7 hours. Therefore, from equation (1.1) and (1.3), we get
$7hr=\dfrac{70km}{{{v}_{a}}-{{v}_{b}}}\Rightarrow {{v}_{a}}-{{v}_{b}}=10\Rightarrow {{v}_{a}}=10+{{v}_{b}}.........(1.4)$
Also, from the question, time taken for them to meet if they travel towards each other =1 hour. Therefore, from equation (1.2) and (1.3), we get
\[\begin{align}
  & 1hr=\dfrac{70km}{{{v}_{a}}+{{v}_{b}}}\Rightarrow {{v}_{a}}+{{v}_{b}}\text{=70}\Rightarrow \text{10+2}{{\text{v}}_{b}}\text{=70 (from equation 1}\text{.4)} \\
 & \Rightarrow {{v}_{b}}=\dfrac{60}{2}km/hr=30km/hr\text{ and }{{\text{v}}_{a}}=10+{{v}_{b}}=40km/hr \\
\end{align}\]
We see that the relative velocities of the cars starting from A and B are 40km/hr and 30km/hr respectively. As these velocities match the option (c), the answer should be option (d).

Note: While calculating the velocities, we got two equations from which we obtained the velocities. We used the method to represent one variable with another to solve the problem. However, one can use other methods of solving linear equations in this question also.