Two vehicles carrying speakers that produce a tone of frequency \(1000.0 \mathrm{~Hz}\) are moving directly toward each other. Vehicle \(\mathrm{A}\) is moving at \(10.00 \mathrm{~m} / \mathrm{s}\) and vehicle \(\mathrm{B}\) is moving at \(20.00 \mathrm{~m} / \mathrm{s}\). Assume the speed of sound in air is \(343.0 \mathrm{~m} / \mathrm{s}\), and find the frequencies that the driver of each vehicle hears.

The general Doppler effect formula is given by: $$f' = \frac{f (v \pm v_o)}{v \pm v_s}$$ where - \(f'\) is the observed frequency, - \(f\) is the emitted/source frequency, - \(v\) is the speed of sound in the medium, - \(v_o\) is the speed of the observer (positive if moving towards the source and negative if moving away from the source), and - \(v_s\) is the speed of the source (positive if moving away from the observer and negative if moving towards the observer). In our case, \(f = 1000.0 \mathrm{~Hz}\), \(v = 343.0 \mathrm{~m} / \mathrm{s}\), \(v_{oA} = -10.00 \mathrm{~m} / \mathrm{s}\) (since driver A is moving towards the source), \(v_{sA} = 20.00 \mathrm{~m} / \mathrm{s}\) (since vehicle B is moving away from driver A), \(v_{oB} = -20.00 \mathrm{~m} / \mathrm{s}\) (since driver B is moving towards the source), and \(v_{sB} = -10.00 \mathrm{~m} / \mathrm{s}\) (since vehicle A is moving towards driver B).

Using the Doppler effect formula, we can find the frequency heard by driver A (\(f_A\)): $$ f_A = \frac{f (v + v_{oA})}{v - v_{sA}}= \frac{1000.0 \mathrm{~Hz} \cdot (343.0 \mathrm{~m} / \mathrm{s} - 10.00 \mathrm{~m} / \mathrm{s})}{343.0 \mathrm{~m} / \mathrm{s} - 20.00 \mathrm{~m} / \mathrm{s}}$$ Solve for \(f_A\): $$ f_A = 1065.52 \mathrm{~Hz} $$

Similarly, we can find the frequency heard by driver B (\(f_B\)): $$ f_B = \frac{f (v + v_{oB})}{v + v_{sB}}= \frac{1000.0 \mathrm{~Hz} \cdot (343.0 \mathrm{~m} / \mathrm{s} - 20.00 \mathrm{~m} / \mathrm{s})}{343.0 \mathrm{~m} / \mathrm{s} + 10.00 \mathrm{~m} / \mathrm{s}}$$ Solve for \(f_B\): $$ f_B = 1193.41 \mathrm{~Hz} $$

The driver of vehicle A will hear a frequency of \(1065.52 \mathrm{~Hz}\), and the driver of vehicle B will hear a frequency of \(1193.41 \mathrm{~Hz}\).