A man is watching two trains,one leaving and | vedclass

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(D) Let $n = 240\,Hz$ be the frequency of the whistles,$v_s = 4\,m/s$ be the speed of the trains,and $v = 320\,m/s$ be the speed of sound.The apparent

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$6$

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(D) Let $n = 240\,Hz$ be the frequency of the whistles,$v_s = 4\,m/s$ be the speed of the trains,and $v = 320\,m/s$ be the speed of sound.The apparent frequency $n'$ heard from the train approaching the observer is given by the Doppler effect formula:$n' = n \left[ \frac{v}{v - v_s} ight]$The apparent frequency $n''$ heard from the train moving away from the observer is given by:$n'' = n \left[ \frac{v}{v + v_s} ight]$The number of beats per second is the difference between these two frequencies:$\Delta n = n' - n'' = n \left[ \frac{v}{v - v_s} - \frac{v}{v + v_s} ight]$$\Delta n = nv \left[ \frac{(v + v_s) - (v - v_s)}{v^2 - v_s^2} ight] = nv \left[ \frac{2v_s}{v^2 - v_s^2} ight]$Since $v_s$ is very small compared to $v$,we can approximate $v^2 - v_s^2 \approx v^2$:$\Delta n \approx nv \left[ \frac{2v_s}{v^2} ight] = \frac{2nv_s}{v}$Substituting the values:$\Delta n = \frac{2 	imes 240 	imes 4}{320} = \frac{1920}{320} = 6\,Hz$Thus,the number of beats per second heard by the man is $6$.

List$-I$	List$-II$
$(P)$ Stephen reaction	$(1)$ Toluene $\xrightarrow{\substack{\text { (i) } CrO _2 Cl _2 / CS _2 \\ \text { (ii) } H _3 O +}}$
$(Q)$ Sandmeyer reaction	$(2)$ Benzoic acid $\xrightarrow{\substack{\text { (i) } PCl _5 \\ \text { (iii) } NH _3 \\ \text { (iii) } P _4 O _{10}, \Delta}}$
$(R)$ Hoffmann bromamide degradation reaction	$(3)$ Nitrobenzene $\xrightarrow{\begin{array}{l}\text { (i) } Fe , HCl \\ \text { (ii) } HCl , NaNO _2 \\ \text { (273-278 K), } H _2 O \end{array}}$
$(S)$ Cannizzaro reaction	$(4)$ Toluene $\xrightarrow{\begin{array}{l}\text { (i) } Cl _2 / hv , H _2 O \\ \text { (ii) } \text {Tollen 's reagent } \\ \text { (iii) } SO _2 Cl _2 \\ \text { (iv) } NH _3\end{array}}$
	$(5)$ Aniline $\xrightarrow{\substack{\text { (i) }\left( CH _3 CO \right)_2 O , \text { Pyridine } \\ \text { (ii) } HN O _3 H _{2} SO _4, 288 K \\ \text { (iii) } aq . NaOH }}$

$A$ man is watching two trains,one leaving and the other coming with equal speed of $4\,m/s$. If they sound their whistles each of frequency $240\,Hz$,the number of beats per second heard by the man will be equal to (velocity of sound in air $= 320\,m/s$)

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