Question

A commuter train passes a passenger platform at a constant speed of 39.6 m/s. The train horn is sounded at its characteristic frequency of 350 Hz. (a) What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding

Answer 1

Complete Question

A commuter train passes a passenger platform at a constant speed of 39.6 m/s. The train horn is sounded at its characteristic frequency of 350 Hz.

(a)

What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding

(b) What wavelength is detected by a person on the platform as the train approaches?

 

Answer:

a

  $\Delta f = 81.93 \ Hz$

b

  $\lambda_1 = 0.867 \ m$

Explanation:

From the question we are told that

      The speed of the train is  $v_t = 39.6 m/s$

      The frequency of the train horn is  $f_t = 350 \ Hz$

Generally the speed of sound has a constant values of  $v_s = 343 m/s$

  Now  according to dopplers equation when the train(source) approaches a person on the platform(observe) then the frequency on the sound observed by the observer can be mathematically represented as  

        $f_1 = f * \frac{v_s}{v_s - v_t}$

substituting values

        $f_1 = 350 * \frac{343 }{343-39.6}$

       $f_1 = 395.7 \ Hz$

  Now  according to dopplers equation when the train(source) moves away from  the  person on the platform(observe) then the frequency on the sound observed by the observer can be mathematically represented as  

           $f_2 = f * \frac{v_s}{v_s +v_t}$

substituting values

        $f_2 = 350 * \frac{343}{343 + 39.6}$

       $f_2 = 313.77 \ Hz$

The overall change in frequency is detected by a person on the platform as the train moves from approaching to receding is mathematically evaluated as

        $\Delta f = f_1 - f_2$

        $\Delta f = 395.7 - 313.77$

        $\Delta f = 81.93 \ Hz$

Generally the wavelength detected by the person as the train approaches  is mathematically represented  as

          $\lambda_1 = \frac{v}{f_1 }$

          $\lambda_1 = \frac{343}{395.7 }$

         $\lambda_1 = 0.867 \ m$