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Difficulty: MediumSound Waves, Echoes, Pitch, Loudness, and Quality

A train moving at a constant speed of 15 m/s15\text{ m/s} towards a tall vertical cliff emits a whistle signal. The driver hears the echo of the whistle 2.0 s2.0\text{ s} after it was sounded. If the speed of sound in air is 340 m/s340\text{ m/s}, what was the distance between the train and the cliff at the instant the whistle was sounded?

  1. 355 m355\text{ m}Answer
  2. B
    340 m340\text{ m}
  3. C
    710 m710\text{ m}
  4. D
    325 m325\text{ m}

Answer

The initial distance between the train and the cliff when the whistle was sounded was 355 m355\text{ m}.
In 2.0 s2.0\text{ s}, the sound wave travels 680 m680\text{ m} while the train advances 30 m30\text{ m} towards the cliff. The total path of the sound consists of the initial distance DD to the cliff plus the return distance (D−30) m(D - 30)\text{ m} to the moving train. Equating D+(D−30)=680 mD + (D - 30) = 680\text{ m} gives 2D=710 m2D = 710\text{ m}, which yields an initial distance of 355 m355\text{ m}.

Step-by-Step Solution

1
Calculate the total distance traveled by the sound wave in the given time interval.
ssound=vsound×t=340 m/s×2.0 s=680 ms_{\text{sound}} = v_{\text{sound}} \times t = 340\text{ m/s} \times 2.0\text{ s} = 680\text{ m}.
Sound travels at a constant speed in air over the total elapsed time of 2.0 s2.0\text{ s}.
2
Calculate the distance covered by the moving train during the same time interval.
strain=vtrain×t=15 m/s×2.0 s=30 ms_{\text{train}} = v_{\text{train}} \times t = 15\text{ m/s} \times 2.0\text{ s} = 30\text{ m}.
The train continues moving towards the cliff while the sound wave travels to the cliff and reflects back.
3
Formulate the geometric path equation for the sound wave.
Let DD be the initial distance to the cliff. Sound travels DD to the cliff and reflects back a distance of (D−30) m(D - 30)\text{ m} to reach the train. Therefore, D+(D−30)=680 mD + (D - 30) = 680\text{ m}.
The echo is received by the driver at a position 30 m30\text{ m} closer to the cliff than where the sound was emitted.
4
Solve the linear equation for DD.
2D−30=680  ⟹  2D=710  ⟹  D=355 m2D - 30 = 680 \implies 2D = 710 \implies D = 355\text{ m}.
Solving for DD gives the exact distance at the moment the whistle was sounded.

Key Concept

Echo path geometry with a moving sound source
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Topics

Sound Waves, Echoes, Pitch, Loudness, and Quality

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