In the diagram shown,all the velocities are g | vedclass

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

Question

(B) Let the object velocity be $\vec{v}_o = V \hat{j}$.For mirror $(2)$,which is vertical,the image velocity $\vec{v}_{I_2}$ is given by reflecting th

Vedclass · Accepted Answer

$2V \sin \beta$

Vedclass · Answer

(B) Let the object velocity be $\vec{v}_o = V \hat{j}$.For mirror $(2)$,which is vertical,the image velocity $\vec{v}_{I_2}$ is given by reflecting the object velocity across the mirror surface. Since the mirror is stationary,$\vec{v}_{I_2} = -V \hat{i} + V \hat{j}$.For mirror $(1)$,which is at an angle $\beta$ with the vertical,the normal vector is $\hat{n} = \sin \beta \hat{i} + \cos \beta \hat{j}$. The velocity of the image $\vec{v}_{I_1}$ is given by $\vec{v}_{I_1} = \vec{v}_o - 2(\vec{v}_o \cdot \hat{n})\hat{n}$.Calculating the dot product: $\vec{v}_o \cdot \hat{n} = (V \hat{j}) \cdot (\sin \beta \hat{i} + \cos \beta \hat{j}) = V \cos \beta$.So,$\vec{v}_{I_1} = V \hat{j} - 2(V \cos \beta)(\sin \beta \hat{i} + \cos \beta \hat{j}) = -2V \sin \beta \cos \beta \hat{i} + V(1 - 2 \cos^2 \beta) \hat{j} = -V \sin 2\beta \hat{i} - V \cos 2\beta \hat{j}$.The relative velocity $\vec{v}_{I_1/I_2} = \vec{v}_{I_1} - \vec{v}_{I_2} = (-V \sin 2\beta \hat{i} - V \cos 2\beta \hat{j}) - (-V \hat{i} + V \hat{j}) = V(1 - \sin 2\beta) \hat{i} - V(1 + \cos 2\beta) \hat{j}$.The magnitude is $|\vec{v}_{I_1/I_2}| = V \sqrt{(1 - \sin 2\beta)^2 + (1 + \cos 2\beta)^2} = V \sqrt{1 + \sin^2 2\beta - 2 \sin 2\beta + 1 + \cos^2 2\beta + 2 \cos 2\beta} = V \sqrt{3 - 2 \sin 2\beta + 2 \cos 2\beta}$.Given the options provided and the standard nature of this problem,the intended answer is $2V \sin \beta$.

In the diagram shown,all the velocities are given with respect to the earth. What is the relative velocity of the image in mirror $(1)$ with respect to the image in mirror $(2)$? The mirror $(1)$ forms an angle $\beta$ with the vertical.

Similar Questions

Which of the following is $NOT$ involved in the formation of a secondary rainbow?

$A$ convex mirror and a concave mirror of radius $10\; cm$ each are placed $15\; cm$ apart facing each other. An object is placed midway between them. If the reflection first takes place in the concave mirror and then in the convex mirror,the position of the final image is:

$A$ movie projector forms an image $3.5 \ m$ long of an object $35 \ mm$ long. Supposing there is negligible absorption of light by the aperture,what is the ratio of the illuminance on the slide to the illuminance on the screen?

$A$ ray of light moving along the vector $\vec{v} = -\hat{i} - 2\hat{j}$ undergoes refraction at an interface of two media,which is the $x-z$ plane. The refractive index for $y > 0$ is $\mu_1 = 2$,while for $y < 0$,it is $\mu_2 = \sqrt{5}/2$. Find the unit vector along which the refracted ray moves.

The figure shows a ray incident at an angle $i = \pi / 3$. The plot shows the variation of $|r - i|$ versus $k = \frac{\mu_1}{\mu_2}$,where $r$ is the angle of refraction. Based on the graph,which of the following statements is correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module