A man in a car at location Q on a straight highway is moving with speed v . He decides to reach a point P in a field at a distance d from highway (point M ) as shown in the figure.Speed of the car in

Question

A man in a car at location Q on a straight highway is moving with speed $v$ . He decides to reach a point $P$ in a field at a distance $d$ from highway (point $M$ ) as shown in the figure.Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach $P$ is minimum?

TestHub · Accepted Answer

Let the car turn of the highway at a distance ' $x$ ' from the point $M$ . So, $RM = x$ And if speed of car in field is $v$ , then time taken by the car to cover the distance $QR =$ $QM - x$ on the highway, $t_{1} = \frac{QM - x}{2 v}$ Time taken to travel the distance ' $R P^{'}$ in the field $t_{2} = \frac{d ^{2} + x ^{2}}{v}$ Total time elapsed to move the car from $Q$ to $P$ $t = t_{1} + t_{2} = \frac{QM - x}{2 v} + \frac{d ^{2} + x ^{2}}{v}$
$or x = \frac{d}{2 ^{2} - 1} = \frac{d}{3}$

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