Question

A source and an observer are each traveling at 0.50 times the speed of sound. The source emits sound waves at \(1.0 \mathrm{kHz}\). Find the observed frequency if (a) the source and observer are moving towand each other, (b) the source and observer are moving away from each other; (c) the source and observer are moving in the same direction.

Short answer

Question: Calculate the observed frequency for each case where the observer is moving with a velocity of 0.50v, and the source is moving with a velocity of 0.50v, by applying the Doppler Effect formula. The original frequency emitted by the source is 1.0 kHz. The cases are as follows: a) The source and observer are moving toward each other. b) The source and observer are moving away from each other. c) The source and observer are moving in the same direction. Answer: a) The observed frequency when the source and observer are moving toward each other is 2 kHz. b) The observed frequency when the source and observer are moving away from each other is 2/3 kHz. c) The observed frequency when the source and observer are moving in the same direction is 1 kHz.

Step-by-step solution

a) Source and observer moving toward each other

In this case, we need to use the plus sign for \(v_{observer}\) and the minus sign for \(v_{source}\). The formula becomes: $$ f_{observed} = \frac{f_{source}(v + v_{observer})}{v - v_{source}} $$ Plug in the values: $$ f_{observed} = \frac{1.0 \mathrm{kHz}(v + 0.50v)}{v - 0.50v} $$ Now, simplify and solve for \(f_{observed}\): $$ f_{observed} = 2 kHz $$ So, for case (a), the observed frequency is \(2 kHz\).

b) Source and observer moving away from each other

In this case, we need to use the minus sign for \(v_{observer}\) and the plus sign for \(v_{source}\). The formula becomes: $$ f_{observed} = \frac{f_{source}(v - v_{observer})}{v + v_{source}} $$ Plug in the values: $$ f_{observed} = \frac{1.0 \mathrm{kHz}(v - 0.50v)}{v + 0.50v} $$ Now, simplify and solve for \(f_{observed}\): $$ f_{observed} = \frac{2}{3} kHz $$ So, for case (b), the observed frequency is \(\frac{2}{3} kHz\).

c) Source and observer moving in the same direction

In this case, we need to use the minus sign for both \(v_{observer}\) and \(v_{source}\). The formula becomes: $$ f_{observed} = \frac{f_{source}(v - v_{observer})}{v - v_{source}} $$ Plug in the values: $$ f_{observed} = \frac{1.0 \mathrm{kHz}(v - 0.50v)}{v - 0.50v} $$ Now, simplify and solve for \(f_{observed}\): $$ f_{observed} = 1 kHz $$ So, for case (c), the observed frequency is \(1 kHz\).