A car approaches you at a constant speed, sounding its horn, and you hear a frequency of 76 Hz. After the car goes by, you hear a frequency of 65 Hz. What is the frequency of the sound emitted by the horn? The speed of sound in air is 343 m/s.

70 Hz

Explanation:

The Doppler equation describes how sound frequency depends on relative velocities:

fr = fs (c + vr) / (c + vs),

where fr is the frequency heard by the receiver,

fs is the frequency emitted at the source,

c is the speed of sound,

vr is the velocity of the receiver,

and vs is the velocity of the source.

Note: vr is positive if the receiver is moving towards the source, negative if away.

Conversely, vs is positive if the receiver is moving away from the source, and negative if towards.

When the car is approaching you:

fr = 76 Hz

vr = 0 m/s

When the car is moving away from you:

fr = 65 Hz

vr = 0 m/s

c, vs, and fs are constant.

We can write two equations:

76 = fs c / (c - vs)

65 = fs c / (c + vs)

If we divide the two equations:

76/65 = [fs c / (c - vs) ] / [fs c / (c + vs) ]

76/65 = [fs c / (c - vs) ] * [ (c + vs) / (fs c) ]

76/65 = (c + vs) / (c - vs)

76 (c - vs) = 65 (c + vs)

76c - 76vs = 65c + 65vs

11c = 141vs

vs = 11/141 c

Substitute into either equation to find fs.

65 = fs c / (c + 11/141 c)

65 = fs c / (152/141 c)

65 = 141/152 fs

fs = 70 Hz