Question

A train standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (a) approaches the platform with a speed of 10 m/s, (b) recedes from the platform with a speed of 10 m/s? (ii) what is the speed of sound in each case? The speed of sound in still air can be taken as 340 m/s

Answer 1

Hint: In this question, we will use the expression for the apparent frequency and by putting the given values we can find the required result. Further, we will see the basic difference between speed and velocity, for our better understanding.

Formula used:
$v' = \left( {\dfrac{v}{{v - {v_r}}}} \right)v$

Complete step by step solution:
a) Here, the frequency of the whistle is given as:
$\nu = 400Hz$

Now, the apparent frequency of the whistle when the train approaches the platform is given as:
$v' = \left( {\dfrac{v}{{v - {v_r}}}} \right)v$

Now, we will put the given values is this expression and solve as:
 $\eqalign{& v' = \left( {\dfrac{{340}}{{340 - 10}}} \right) \times 400 \cr
  & \Rightarrow v' = 412.12Hz \cr} $

b) Now, the apparent frequency (v′’) of the whistle when the train recedes from the platform is given as:
$v'' = \left( {\dfrac{v}{{v + {v_r}}}} \right)v$

Now, we will put the given values is this expression and solve as:
$\eqalign{& v'' = \left( {\dfrac{{340}}{{340 + 10}}} \right) \times 400 \cr
  & \Rightarrow v'' = 388.57Hz \cr} $

As we know that the apparent change in the frequency of sound is observed due to the relative motions of the given source and the observer. Now, these relative motions produce no effect on the speed of sound.

So, the speed of sound in the air in both the given cases remains the same, i.e., given by 340 m/s.

Additional information:
We already know the basic difference of speed and velocity i.e., speed is the measure of how fast an object can travel, whereas velocity gives us the direction of this speed. Speed is a scalar quantity that means it has only magnitude, whereas velocity is a vector quantity that means it has both magnitude and direction. The S.I unit of velocity is meter per second (m/sec).

Note:
We should remember that the speed of sound is maximum in solid medium. Since, we know speed does not have any direction therefore, speed cannot be negative. We must remember that the unit of speed and velocity is the same i.e., meter per sec or m/s.