Question

Two cars are moving in the same direction with the same speed $30\mathrm{km}/\mathrm{hr}$. They are separated by distance of $5\mathrm{km}$, the speed of a car moving in the opposite direction if it meets these two cars at an interval of $4\min$, will be

• A. $40\mathrm{km}/\mathrm{hr}$
• B. $45\mathrm{km}/\mathrm{hr}$
• C. $30\mathrm{km}/\mathrm{hr}$
• D. $15\mathrm{km}/\mathrm{hr}$

Answer 1

The correct option is B

$45\mathrm{km}/\mathrm{hr}$

Step 1. Given data

Speed of two cars are $30\mathrm{km}/\mathrm{hr}$, distance between two car is $5\mathrm{km}$and time interval $\mathrm{t}$ is $4\min$.

We have to find out the actual velocity of third car.

Step 2. Formula to be used

The relative velocity can be calculated by,

${\overrightarrow{\mathrm{V}}}_{\mathrm{AB}}={\overrightarrow{\mathrm{V}}}_{\mathrm{A}}-{\overrightarrow{\mathrm{V}}}_{\mathrm{B}}$

The formula for speed is,

$\text{speed}={\displaystyle \frac{\text{distance}}{\text{time}}}$.
Step 3. Find the velocity of third car

Let,

The velocity of the first two cars be ${\mathrm{v}}_1=30\mathrm{km}/\mathrm{hr}$ and ${\mathrm{v}}_2=30\mathrm{km}/\mathrm{hr}$, the distance between them be $\mathrm{d}=5\mathrm{km}$
So, the velocities of the two cars are the equal, the relative speed between them is $0$, which means, for a person in car $2$, car $1$ seems stationary.

It implies that the distance between car $1$ and car $2$ is $\mathrm{d}=5\mathrm{km}$ via the motion.
Consider that car 3 has a velocity $=-\mathrm{xkm}/\mathrm{hr}$, thus it is moving in the opposite direction as of car $1$ and car $2$.
We will take car $2$ as the reference frame.
From this perspective, it appears like two points (car $1$ and car $2$) are separated by distance $\mathrm{d}$ on a $1-$dimensional plane. Car $3$ is now getting close to the points.

A person in car $2$ will observe that car $3$ is moving at the following speed,

${\overrightarrow{\mathrm{V}}}_{32}={\overrightarrow{\mathrm{V}}}_3-{\overrightarrow{\mathrm{V}}}_2$

$=-\mathrm{x}-30$

$=-\left(\mathrm{x}+30\right)\mathrm{km}/\mathrm{hr}$

Step 4. Calculate the time taken by third car
From the given, time taken by car $3$ to cross the other cars is,

$\mathrm{t}=4\min$

$=\frac{4}{60}\mathrm{hr}$

$=\frac{1}{15}\mathrm{hr}$

Step 5. Find the speed
Let's only examine speed and distance since we are dealing with a $1-$dimensional situation.
Use the formula for speed, we get

$\text{speed}={\displaystyle \frac{\text{distance}}{\text{time}}}$

$30+\mathrm{x}=\frac{5}{{\displaystyle \frac{1}{15}}}$

$30+\mathrm{x}=75\mathrm{km}/\mathrm{hr}$

$\mathrm{x}=75-30\mathrm{km}/\mathrm{hr}$

$\mathrm{x}=45\mathrm{km}/\mathrm{hr}$
Therefore, option $\mathrm{B}$ is the correct answer