A man in a car at location Q on a straight hi | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the car turn off the highway at a distance $x$ from point $M$. So,$RM = x$.Let the speed of the car on the highway be $v_h = v$ and the speed

Vedclass · Accepted Answer

$\frac{d}{\sqrt{3}}$

Vedclass · Answer

(A) Let the car turn off the highway at a distance $x$ from point $M$. So,$RM = x$.Let the speed of the car on the highway be $v_h = v$ and the speed in the field be $v_f = v/2$.The distance $QM$ is constant. Let $QM = L$. The distance covered on the highway is $QM - x = L - x$.The time taken to travel on the highway is $t_1 = \frac{L - x}{v}$.The distance $RP$ in the field is $\sqrt{d^2 + x^2}$.The time taken to travel in the field is $t_2 = \frac{\sqrt{d^2 + x^2}}{v/2} = \frac{2\sqrt{d^2 + x^2}}{v}$.Total time $t = t_1 + t_2 = \frac{L - x}{v} + \frac{2\sqrt{d^2 + x^2}}{v}$.For minimum time,$\frac{dt}{dx} = 0$.$\frac{d}{dx} \left( \frac{L - x}{v} + \frac{2\sqrt{d^2 + x^2}}{v} \right) = 0$.$\frac{1}{v} \left( -1 + 2 \cdot \frac{1}{2\sqrt{d^2 + x^2}} \cdot 2x \right) = 0$.$-1 + \frac{2x}{\sqrt{d^2 + x^2}} = 0 \implies \frac{2x}{\sqrt{d^2 + x^2}} = 1$.$4x^2 = d^2 + x^2 \implies 3x^2 = d^2 \implies x = \frac{d}{\sqrt{3}}$.

Similar Questions

$A$ street car moves rectilinearly from station $A$ to the next station $B$ with an acceleration varying according to the law $a = (b - cx)$,where $b$ and $c$ are constants and $x$ is the distance from station $A$. The distance between the two stations and the maximum velocity are:

$A$ train starting from rest first accelerates uniformly up to a speed of $80 \ km/h$ for time $t$,then it moves with a constant speed for time $3t$. The average speed of the train for this duration of journey will be (in $km/h$):

$A$ point traversed half of the distance with a velocity $v_0$. The remaining part of the distance was covered with velocity $v_1$ for half the time and with velocity $v_2$ for the other half of the time. The mean velocity of the point averaged over the whole time of motion is

Vedclass Test Series

Exam Paper Generator

Online Exam Module