A hunter is standing on flat ground between two vertical...

A hunter is standing on flat ground between two vertical cliffs that are directly opposite one another. He is closer to one cliff than to the other. He fires a gun and, after a while, hears three echoes. The second echo arrives $1.6 \mathrm{~s}$ after the first, and the third echo arrives 1.1 s after the second. Assuming that the speed of sound is $343 \mathrm{~m} / \mathrm{s}$ and that there are no reflections of sound from the ground, find the distance between the cliffs.

Key Concepts

Algebraic Modeling and System of Equations

When multiple echoes are observed and the corresponding time intervals differ, setting up a system of equations based on the distance traveled by the sound for each echo is essential. Solving these equations using algebra helps determine the unknown distances in the problem, such as the separation between reflecting surfaces.

Time-of-Flight Analysis

Time-of-flight analysis involves using the time delay between the emission of a sound and the detection of its echo to calculate distances. By knowing the sound speed, one can relate these delays to the round-trip distance the sound waves travel, a fundamental technique in wave-based measurement systems.

Geometric Considerations in Reflection

Sound reflection problems often require analyzing the geometry of the setup, including the relative positions of the sound source and reflecting surfaces. The distances are typically represented as the sum or difference of multiple path segments, which are then used to derive relationships between the echoes.

Speed of Sound

In acoustics, the speed of sound specifies the rate at which sound travels through a medium. It is crucial for converting the time intervals between the emission of a sound and the reception of its echoes into the distances traveled, forming the basis for many time-of-flight calculations.

Sound Reflection and Echoes

Echoes are the result of sound waves reflecting off obstacles. Understanding how sound reflects off surfaces, such as cliffs, is key to interpreting delays between successive echoes as twice the distance from the emitter to a reflective surface, helping determine spatial separations.

Transcript

00:01 In this problem, we have to calculate the distance between the cliffs, which we call d.

00:07 Let's calculate the distance from the hunter to the nearest cliff through the expression do you want equals? two.

00:14 We do you want you wanted by to their school record in one most holding 41 we can write.

00:20 It is, do you want to close to do you want? do you want it by? well, let's call it equation, too.

00:28 The distance from the hunter to that seven cliff is d two equals two.

00:34 Greedy, too divided by two quality.

00:37 Question three.

00:39 Here do you do is the time after which second ego is hurt.

00:45 The second equities is generated by the far cliff.

00:50 Now the time interval between first and second equal his routine as delta t equals two d two minus t.

00:58 One.

00:59 No, it can't be lit on 40 tewes, do you two equals 2 to 1 plus delta t.

01:05 This is a question number four a 30 could be heard in a time off after a time off.

01:12 Deeply, this tree tree tree will be close to the one you wanted by b.

01:19 The sequel is generated by the reflection off the circuit.

01:22 Nickel from the nearest wall.

01:24 So we call this equation with number five? no.

01:28 Big comparing.

01:29 Equating number two with equity number five...

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