Considering the set – up shown, choose the correct alternatives considering rays to be approximately paraxial. (Height of butterfly above the principal axis is 1m and all velocities are relative to ground)

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Velocity of butterfly’s image (direct reflection) in mirror is $- 15 \hat{i} + 11 \hat{j}$ (m/s).

Velocity of butterfly’s image (direct reflection) w.r.t. butterfly is at angle $\tan^{- 1} \frac{1}{3}$ w.r.t horizontal.

Velocity of butterfly w.r.t. fish is parallel to butterfly’s image’s velocity in mirror (direct reflection) if velocity of fish is $7.5 \hat{i} + 2.5 \hat{j}$ (m/s).

Velocity of butterfly’s (direct reflection) image in mirror is $- 12 \hat{i} + 11 \hat{j}$ (m/s).

answer is A, B, C.

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Detailed Solution

(a) ${\vec{V}}_{{(I / m)}_{11}} = - m^{2} {\vec{V}}_{{(O / m)}_{11}} m = \frac{f}{f - u}$
$\frac{y_{I}}{y_{o}} = \frac{f}{f - u}, {\vec{V}}_{(I / m) ⊥} = \frac{d y_{I}}{d t}$

y=position of image or object measured perpendicular to principal axes
Velocity of image with respect to ground, $\vec{V_{I}} = {\vec{V}}_{{(I / m)}_{11}} + {\vec{V}}_{(I / m) ⊥} + \vec{V_{m}}$
(B) Velocity of butterfly with respect to fish, ${\vec{V}}_{B F} = (6 \times \frac{4}{3}) \hat{j} - {\vec{V}}_{F}$