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Question

# A small musical toy produces a steady tone at $1000 \mathrm{~Hz}$. It is accidentally dropped from the window of a tall apartment building. How far has it fallen if the frequency heard from the window is $950 \mathrm{~Hz}$ ? (Ignore air friction, and use $340 \mathrm{~m} / \mathrm{s}$ for the speed of sound.)

Solution

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From equation 12.19, i.e. the master doppler effect equation,

$f_L = \frac{v+ v_L}{v+v_S} f_S \hspace{0.5 cm} (1)$

Where $f_L$ is the frequency of the sound heard by the listener, $f_S$ is the frequency of sound produced by the source, $v$ is the velocity of sound, $v_L$ and $v_S$ are the velocities of the listener and the source respectively.

Equation 2.14 gives us the relationship between an object's height and velocity as,

$v^2 = v_0 ^2 - 2g(y-y_0) \hspace{0.5 cm} (2)$

Where $v_0$ is the intial velocity, $y$ and $y_0$ are the final and initial distances from the origin, and $g$ is the acceleration due to gravity.

Given: $f_S = 1000$ Hz, $f_L = 950$ Hz, $v = 340$ m/s, $v_L = 0$, $v_0 = 0$, $y = 0$

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