Question

Two cars are moving in the same directions with a speed of 30km/h. They are separated from each other by 5km. Third car moving in the opposite direction meets the two cars after an interval of 4 minutes. What is the speed of the third car?
a)30km/h
b)35km/h
c)40km/h
d)45km/h

Answer 1

Hint: In the above question it is given to us that there are two cars moving with the same speed. Hence at any point of time they will observe each other to be at the same distance apart. Further it is given to us that a third car crosses them at an interval of 4min. Therefore we will use the concept of relative velocity between the car moving opposite to the other two and the car which is behind the other car where they are both moving with the same speed. Using this concept we will obtain a relation between the distance of separation and time and speed of the car moving opposite and accordingly we will obtain the required answer.
Formula used:
$V(rel)=\dfrac{D}{t}$
$V(rel)={{v}_{1}}-{{v}_{2}}$

Complete answer:
In the above question it is given to us that the two cars i.e. A and B are moving with the same velocity of 30km/h. Let us say the car A is moving ahead of car B. Therefore we can imply that at any given instant of time, both these cars will be at the same distance apart i.e. 5km.
Now in the question it is given to us that a third car i.e. C moves opposite to them such that it crosses the two at an interval of 4min. Let us say at time t=0, the car C just reaches the car A. Now at this point the distance between the car C and B is D=5km and it meets the car B after t=4min. Now this distance will be covered by the two cars as per the relative velocity of the two. Let us say the velocity of car B is ${{v}_{1}}$ and that of C be ${{v}_{2}}$. Therefore the relative velocity ($V(rel)$) of the two is given by,
$V(rel)={{v}_{1}}-{{v}_{2}}$. Now we know that the two cars are moving towards each other such that they are separated by finite distance. Hence the distance covered in time t can be written as,
$\begin{align}
  & V(rel)=\dfrac{D}{t} \\
 & \Rightarrow {{v}_{1}}-{{v}_{2}}=\dfrac{5km}{4\min }=\dfrac{60\times 5km}{4}=75km/h \\
 & \Rightarrow 30km/h-{{v}_{2}}=75km/h \\
 & \Rightarrow {{v}_{2}}=-45km/h \\
\end{align}$

So, the correct answer is “Option D”.

Note:
The minus sign indicates that the car C moves opposite to that of car B as well as that of A. In the solution we have taken the relative velocity as the difference between car B and car C. It is to be noted that this can also be done vice versa but then the sign convention will accordingly alter.