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Ranvir and Akshay are rowing two boats in a river. with the same speed v relative to the river. Ranvir is upstream relative to Akshay. The river is flowing with speed v/2. Each boat sends, through the water, a signal to the other. The frequencies $f_{0}$ of the generated continuous sonic signals are the same. Each observer observes the wave sent by the other person. In each of the situation in column-I, match the appropriate entries given in column-II.
Column I Column II
a)

p) The times it takes for the signals to travel from one boat to the other will be the same.
b)
q) The wavelength greater than the wavelength observe by Ranvir.
c)
r) The frequency observed by Akshay is greater than the frequency observed by Ranvir
d)
s) The wavelength observed by Akshay is lesser than the wavelength observed by Ranvir
t) The frequency observed by Akshay is equal to the frequency observed by Ranvir

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$[a \to q, s; b \to p, s; c \to r, t; d \to r]$

$[a \to q, s, t; b \to s; c \to p, r; d \to s]$

$[a \to p, t; b \to p, t; c \to r; d \to p, r]$

$[a \to q, t; b \to q, r; c \to s, t; d \to q]$

answer is A.

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Detailed Solution

$[a \to q, t; b \to q, r; c \to s; d \to q]$

Time taken by wave to travel from Ranvir to Akshay
$T_{A} = \frac{d}{C + \frac{v}{2} + v}; T_{B} = \frac{d}{C + \frac{v}{2} - v}; T_{C} = \frac{d}{C + \frac{v}{2} - v}; T_{D} = \frac{d}{C + \frac{v}{2} + v};$

Time taken by wave to travel from akshay to Ranvir
$T_{A} = \frac{d}{C - \frac{v}{2} - v}; T_{B} = \frac{d}{C - \frac{v}{2} - v}; T_{C} = \frac{d}{C - \frac{v}{2} + v}; T_{D} = \frac{d}{C - \frac{v}{2} + v};$

Time will not be same for any case.

Wavelength observed by Akshay
$λ_{A} = \frac{C + \frac{v}{2} + v}{f}; λ_{B} = \frac{C + \frac{v}{2} + v}{f}; λ_{C} = \frac{C + \frac{v}{2} - v}{f}; λ_{D} = \frac{C + \frac{v}{2} - v}{f};$

Wavelength observed by Ranvir

$λ_{A} = \frac{C - \frac{v}{2} - v}{f}; λ_{B} = \frac{C - \frac{v}{2} + v}{f}; λ_{C} = \frac{C - \frac{v}{2} + v}{f}; λ_{D} = \frac{C - \frac{v}{2} - v}{f};$

Frequency observed by Akshay

$f_{A} = f; f_{B} = f [\frac{c + \frac{v}{2} - v}{c + \frac{v}{2} + v}]; f_{C} = f; f_{D} = f [\frac{C + \frac{v}{2} + v}{C + \frac{v}{2} - v}]$

Frequency observed by Akshay

$f_{A} = f; f_{B} = f [\frac{c - \frac{v}{2} - v}{c - \frac{v}{2} + v}]; f_{C} = f; f_{D} = f [\frac{C - \frac{v}{2} + v}{C - \frac{v}{2} - v}]$

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