An insect starts moving up in a liquid from p | vedclass

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

Question

(D) The apparent depth $h$ of an object at a depth $y$ in a medium with variable refractive index $\mu(y)$ is given by the integral $h = \int_{0}^{y}

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) The apparent depth $h$ of an object at a depth $y$ in a medium with variable refractive index $\mu(y)$ is given by the integral $h = \int_{0}^{y} \frac{dy'}{\mu(y')}$.Given $\mu = \mu_0(1 + ay)$,the apparent depth $h$ is:$h = \int_{0}^{y} \frac{dy'}{\mu_0(1 + ay')} = \frac{1}{\mu_0 a} [\ln(1 + ay')]_{0}^{y} = \frac{\ln(1 + ay)}{\mu_0 a}$.The apparent speed $v_{app}$ is the rate of change of apparent depth with respect to time:$v_{app} = \frac{dh}{dt} = \frac{d}{dt} \left( \frac{\ln(1 + ay)}{\mu_0 a} \right)$.Using the chain rule,$\frac{dh}{dt} = \frac{1}{\mu_0 a} \cdot \frac{1}{1 + ay} \cdot \frac{d(1 + ay)}{dt} = \frac{1}{\mu_0 a} \cdot \frac{1}{1 + ay} \cdot (a \frac{dy}{dt})$.Since the speed of the insect is $u = -\frac{dy}{dt}$ (as $y$ decreases as it moves up),we have $\frac{dy}{dt} = -u$.Thus,the magnitude of the apparent speed is $v_{app} = \frac{u}{\mu_0(1 + ay)}$.Comparing this with the given options,none of the expressions match the derived result exactly as a constant,as it depends on the position $y$. Therefore,the correct choice is 'None of these'.

An insect starts moving up in a liquid from point $O$ with a variable refractive index $\mu = \mu_0(1 + ay)$,where $y$ is the depth of the liquid from the surface. If $u$ is the speed of the insect,what is its apparent speed to the observer $E$?

Similar Questions

Calculate the value of $n_{32} \times n_{21} = .... $

$A$ fish in water (refractive index $n$) looks at a bird vertically above in the air. If $y$ is the height of the bird and $x$ is the depth of the fish from the surface,then the distance of the bird as estimated by the fish is

The length of the optical path of two media in contact of lengths $d_1$ and $d_2$ with refractive indices $\mu_1$ and $\mu_2$ respectively,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module