Question

A car $M$ right now on the cross road is moving towards the west at $40\,kmh{r^{ - 1}}$. Another car $P$ which is right now at $5\,km$ south of the crossing is moving towards it at $40\,kmh{r^{ - 1}}$. The closest distance of approach between the two cars will be:
(A) $5\,km$
(B) $2.5\,km$
(C) $5\sqrt 2 \,km$
(D) $\dfrac{5}{{\sqrt 2 }}\,km$

Answer 1

Hint: The closest distance between the two cars can be determined by using the trigonometry equation because the given information in the question forms the triangle. By using the speed of the car, the angle can be determined. By using that angle the distance can be determined.

Complete step by step solution
Given that,
The speed of the car $M$ is, $40\,kmh{r^{ - 1}}$,
The distance between the car $P$ and the cross road is, $5\,km$,
The speed of the car $P$ is, $40\,kmh{r^{ - 1}}$.

Now, by using the velocities, then
$\tan \theta = \dfrac{{{v_1}}}{{{v_2}}}$
Where, ${v_1}$ is the velocity of the car $M$ and ${v_2}$ is the velocity of the car $P$.
By substituting the velocity of the car $M$ and velocity of the car $P$ in the above equation, then
$\tan \theta = \dfrac{{40}}{{40}}$
By dividing the terms in the above equation, then the above equation is written as,
$\tan \theta = 1$
By rearranging the terms in the above equation, then the above equation is written as,
$\theta = {\tan ^{ - 1}}\left( 1 \right)$
From the trigonometry, the value of the ${\tan ^{ - 1}}\left( 1 \right) = {45^ \circ }$, substitute this value in the above equation, then
$\theta = {45^ \circ }$
From the triangle the angle between is ${45^ \circ }$, by using this angle the distance $d$ can be determined.
Now, using the angle and the distance values, then
$\sin \theta = \dfrac{{5\,km}}{d}$
By substituting the angle value in the above equation, then
$\sin {45^ \circ } = \dfrac{{5\,km}}{d}$
By rearranging the terms in the above equation, then
$d = \dfrac{{5\,km}}{{\sin {{45}^ \circ }}}$
From the trigonometry, the value of the $\sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}$, substitute this value in the above equation, then
$d = \dfrac{{5\,km}}{{\left( {\dfrac{1}{{\sqrt 2 }}} \right)}}$
By rearranging the terms in the above equation, then
$d = 5\sqrt 2 \,km$

Hence, the option (C) is the correct answer.

Note: From the given information, the triangle is formed and then by using the velocities of the two cars of $M$ and $P$, the angle between the two cars can be determined and then by using the angle values, and the distance given in the question, then the distance between the two cars can be determined.