Question

a speaker (sound source) is located at the origin, one pickup microphone is located at distance D1 away and along the positive y-axis and a second pick up microphone is located a distance d=2.47m to the right of the first microphone at a distance D2 from the speaker. If a sound from the speaker reaches the second microphone 2.54 ms after it reaches the first microphone and the speed of sound is 343 m/s, determine the distances D1 and D2 to the microphone

Answer 1

Question #78090, Physics / Other

A speaker (sound source) is located at the origin, one pickup microphone is located at distance D1 away and along the positive y-axis and a second pick up microphone is located at a distance $d = 2.47m$ to the right of the first microphone at a distance D2 from the speaker. If a sound from the speaker reaches the second microphone 2.54 ms after it reaches the first microphone and the speed of sound is $343~m/s$, determine the distances D1 and D2 to the microphone

Solution

1)

$$\frac{D_2}{v} - \frac{D_1}{v} = \frac{D_2 - D_1}{v} = t$$ $$D_2 - D_1 = vt$$

2)

$$D^2 = D_2^2 - D_1^2 = (D_2 - D_1)(D_2 + D_1) = vt(D_2 + D_1)$$ $$\frac{D^2}{vt} = D_2 + D_1 = D_1 + vt + D_1$$ $$D_1 = \frac{1}{2} \left( \frac{D^2}{vt} - vt \right) = \frac{1}{2} \left( \frac{2.47^2}{(343 \cdot 0.00254)} - (343 \cdot 0.00254) \right) = 3.07~m.$$ $$D_2 = \frac{1}{2} \left( \frac{D^2}{vt} - vt \right) + vt = \frac{1}{2} \left( \frac{D^2}{vt} + vt \right) = \frac{1}{2} \left( \frac{2.47^2}{(343 \cdot 0.00254)} + (343 \cdot 0.00254) \right) = 3.94~m.$$

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