Question

Two cars $\text{A}$ and $\text{B}$ are moving with uniform speeds as shown in the figure given below, at $t=0$. Find out the shortest distance between the two cars.

• A. $2\sqrt{5}\text{m}$
• B. $4\sqrt{5}\text{m}$
• C. $8\sqrt{5}\text{m}$
• D. $6\sqrt{5}\text{m}$

Answer 1

The correct option is B $4\sqrt{5}\text{m}$

First, let us find the velocity of $\text{A}$ w.r.t $\text{B}$ in the given problem.

Let $\theta$ be the angle made by the $\overrightarrow{V_{AB}}$with the $\overrightarrow{-V_B}$


Using the figure, we get

$\tan \theta =|\frac{15}{30}|=\frac{1}{2}$

Also $\tan \theta =\frac{AD}{CD}=\frac{AD}{40}=\frac{1}{2}$

$\Rightarrow$$AD=20\text{m}$

$\Rightarrow$$DO=30-20=10\text{m}$

$\Rightarrow$$BC=10\text{m}$

In $\Delta BCE$

$d_{unit}=BC\cos \theta =10\cos \theta$

$\Rightarrow d_{unit}=\frac{10\times 2}{\sqrt{5}}=4\sqrt{5}\text{m}$

Since closest distance is non-zero, so they will not collide.

Hence, option (b) is the correct answer.

Why this question ? To find the shortest distance between $\text{A}$ and $\text{B}$ whose relative acceleration is zero, 1) Draw a line along relative velocity of a w.r.t $B$ 2) Draw a perpendicular from $B$ on this line 3) Length of this perpendicular will give the shortest distance.