Question

Jane waits on a railroad platform while two trains approach from the same direction at equal speeds of 7.10 m/s. Both trains are blowing their whistles (which have the same frequency), and one train is some distance behind the other. After the first train passes Jane but before the second train passes her, she hears beats of frequency 6.50 Hz. What is the frequency of the train whistles

Answer 1

Answer:

156.96691 Hz

Explanation:

f' = Actual frequency

v = Velocity of sound in air = 343 m/s

$v_r$ = Relative velocity of train = 7.1 m/s

$\Delta f$ = Beat frequency = 6.5 Hz

Receding frequency

$f_1=f'\dfrac{v}{v-v_r}\\\Rightarrow f_1=f'\dfrac{343}{343-7.1}\\\Rightarrow f_1=f'1.02113$

Approaching frequency

$f_2=f'\dfrac{v}{v+v_r}\\\Rightarrow f_2=f'\dfrac{343}{343+7.1}\\\Rightarrow f_2=f'0.97972$

Beat frequency is given by

$\Delta f=f_1-f_2\\\Rightarrow 6.5=f'1.02113-f'0.97972\\\Rightarrow 6.5=f'0.04141\\\Rightarrow f'=\dfrac{6.5}{0.04141}\\\Rightarrow f'=156.96691\ Hz$

Actual frequency of the train whistle is 156.96691 Hz