Physics Mystery: How Far Has the Tuning Fork Fallen?

Question:

A physics student drops a vibrating 440-hz tuning fork down the elevator shaft of a tall building. When the student hears a frequency of 400 hz, how far has the tuning fork fallen?

Answer:

1. The speed of the source is 34m/s, using the Doppler effect.
2. The location of the source at that time (t₁) is 3.466s.
3. The time (t₂) for sound to travel 58.92m is 0.173s.
4. The distance the source has fallen in time (t₁ + t₂) is 65m.

The Doppler effect, also known as the Doppler shift, is a phenomenon where a wave's frequency changes in response to an observer moving in respect to the wave source. This effect was first reported by Austrian physicist Christian Doppler in 1842.

Mathematically, the Doppler effect is written as: f' = ((v + v₀) / (v - vₛ)) * f, where f' is the perceived frequency, v is the speed of sound, v₀ is the speed of the observer, vₛ is the speed of the source, and f is the actual frequency of the source.

1. Using the Doppler effect formula, we find that the speed of the source is 34m/s.

2. By calculating the distance fallen using the speed obtained from the Doppler effect, we find that the location of the source at time t₁ is 3.466s.

3. The time taken for sound to travel 58.92m is found to be 0.173s.

4. Using the formula for distance fallen due to gravity, we determine that the distance the source has fallen in time (t₁ + t₂) is 65m.

Therefore, the physics student has dropped the tuning fork a distance of 65m down the elevator shaft of the tall building.