A train S1, moving with a uniform velocity of 108 km/h approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2 as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is _____

[Speed of the sound = 330 m/s] [2019]

Question

A train S1, moving with a uniform velocity of 108 km/h approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2 as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is _____

[Speed of the sound = 330 m/s] [2019]

Smart Achievers · Accepted Answer

(8.13)

$Apparent frequency heard by observer due to source S_{1}$

$ν_{1} = ν [\frac{v + v_{0}}{v}] = 120 [\frac{330 + 10}{330}] = 120 \times \frac{34}{33} = 123.636 Hz$

$Apparent frequency heard by observer due to source S_{2}$

$ν_{2} = ν [\frac{v + v_{0}}{v - v_{s}}] = 120 [\frac{330 + 10 \cos 53 °}{330 - 30 \cos 37 °}]$

$∴$ $ν_{2} = 120 [\frac{330 + 10 \times 0.6}{330 - 30 \times 0.8}] = 120 [\frac{336}{306}] = 131.764 Hz$

$∴$ $Beat frequency, ν_{b} = ν_{2} - ν_{1} = 131.764 - 123.636 = 8.125 Hz$ $\approx 8.13 Hz$

Solution

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