Answer
(a) The observed frequency is $3.0~kHz$
(b) The observed frequency is $333.3~Hz$
(c) The observed frequency is $1.0~kHz$
Work Step by Step
(a) We can use the equation for the Doppler effect when the source and observer are moving toward each other:
$f_o = \left(\frac{v+v_o}{v-v_s}\right)~f_s$
$f_o = \left(\frac{v+v/2}{v-v/2}\right)~f_s$
$f_o = \left(\frac{3v/2}{v/2}\right)~f_s$
$f_o = 3~f_s$
$f_o = (3)~(1.0~kHz)$
$f_o = 3.0~kHz$
The observed frequency is $3.0~kHz$
(b) We can use the equation for the Doppler effect when the source and observer are moving away from each other:
$f_o = \left(\frac{v-v_o}{v+v_s}\right)~f_s$
$f_o = \left(\frac{v-v/2}{v+v/2}\right)~f_s$
$f_o = \left(\frac{v/2}{3v/2}\right)~f_s$
$f_o = \left(\frac{1}{3}\right)~f_s$
$f_o = \left(\frac{1}{3}\right)~(1000~Hz)$
$f_o = 333.3~Hz$
The observed frequency is $333.3~Hz$
(c) We can use the equation for the Doppler effect when the source and observer are moving in the same direction:
$f_o = \left(\frac{v-v_o}{v-v_s}\right)~f_s$
$f_o = \left(\frac{v-v/2}{v-v/2}\right)~f_s$
$f_o = f_s$
$f_o = 1.0~kHz$
The observed frequency is $1.0~kHz$.
