Question

A man in a car at location Q on a straight highway is moving with speed $\upsilon$. He decides to reach a point P in a field at a distance d from the 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?

Answer 1

Given speed of the vehicle in highway= V m/s and its speed in field =$\frac{v}{2}$ m/s

Time taken to travel from R to P $=\frac{RP}{V_{infield}}=\frac{2\sqrt{{(z-x)}^2+d^2}}{v}$

The value of RP is found by using Pythagoras theorem.

Now total time taken to travel from point Q to point P taking diversion from R

$t_{total}=\frac{x}{v}+\frac{2\sqrt{{(z-x}^2)+d^2}}{v}$

For the time T to minimum, $\frac{dT}{dx}=0$

$\frac{1}{v}+\frac{2}{v}\frac{1}{2\sqrt{d^2+{(z-x)}^2}}(-2)(z-x)=0(-2\left)(z-x\right)=\sqrt{d^2+{(z-x)}^2}4(z-x{)}^2=d^2+{(z-x)}^{2}z-x=\frac{d}{\sqrt{3}}$