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 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 ?

• A. $d$
• B. $\frac{d}{\sqrt{2}}$
• C. $\frac{d}{{2}}$
• D. $\frac{d}{\sqrt{3}}$

Answer 1

The Correct Option is D

1. To solve the problem, we must find the distance $RM$ such that the time taken to reach point $P$ is minimized.
2. The path consists of two segments:
   • Movement along the highway from $Q$ to $M$ (distance $RM$) with speed $v$.
   • Movement through the field from $M$ to $P$ (distance $\sqrt{d^2 + (PM)^2}$) with speed $v/2$ (since speed through field is half).
3. The objective is to minimize the total time, $T$, required to reach $P$. Thus, $$ T = \frac{RM}{v} + \frac{\sqrt{d^2 + (PM)^2}}{v/2} $$.
4. Substituting $PM = RM - RM$ gives: $$ T = \frac{RM}{v} + \frac{\sqrt{d^2 + (d - RM)^2}}{v/2} $$.
5. To minimize $T$, find $dT/d(RM) = 0$. This involves taking the derivative: $$ \frac{d}{d(RM)} \left( \frac{RM}{v} + \frac{2\sqrt{d^2 + (d - RM)^2}}{v} \right) = 0 $$.
6. The critical step involves simplifying and solving the derivative, but this elaborates into solving: $$ \frac{1}{v} + \frac{-2(d - RM)}{v\sqrt{d^2 + (d - RM)^2}} = 0 $$.
7. This simplifies to: $$ (d - RM)^2 = d^2/3 $$ which gives: $$ RM = d/\sqrt{3} $$.
8. Thus, the distance $RM$ to minimize the time to reach $P$ is $d/\sqrt{3}$, making the correct answer to the question the second option.