Two thin metal rods are arranged in water parallel to each other a distance d apart. Speeds c and $ν$ of sound in the water and in the rod respectively are not equal. If one of the rods is tapped at point A, the shortest time the sound takes to move from the point A to a point B on the other rod.

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$t = \frac{\sqrt{d^{2} + l^{2}}}{c} i f \frac{c}{ν} \geq \frac{l}{\sqrt{d^{2} + l^{2}}}$

$t = \frac{l}{v} + d \sqrt{\frac{1}{c^{2}} - \frac{1}{v^{2}}} i f \frac{c}{ν} < \frac{l}{\sqrt{d^{2} + l^{2}}}$

$t = \frac{\sqrt{d^{2} + l^{2}}}{c} i f \frac{c}{ν} \leq \frac{l}{\sqrt{d^{2} + l^{2}}}$

$t = \frac{l}{v} + d \sqrt{\frac{1}{c^{2}} - \frac{1}{v^{2}}} i f \frac{c}{v} > \frac{l}{\sqrt{d^{2} + l^{2}}}$

answer is A, B.

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Detailed Solution

$λ > \frac{3}{2}$
$i f \sin θ_{c} > \frac{ρ}{\sqrt{l^{2} + d^{2}}}$
then it travels fully inside the water medium
$t = \frac{\sqrt{d^{2} + l^{2}}}{c}$
If $\sin θ_{C} = \frac{l}{\sqrt{l^{2} + d^{2}}}$ then it travels partly in bar & partly in water
$t = \frac{x}{v} + \frac{\sqrt{{(l - x)}^{2} + d^{2}}}{c}$
$l - x = \frac{d c}{\sqrt{v^{2} - c^{2}}}$