   Chapter 14, Problem 32P

Chapter
Section
Textbook Problem

A bat flying at 5.00 m/s is chasing an insect flying in the same direction. If the bat emits a 40.0-kHz chirp and receives back an echo at 40.4 kHz, (a) what is the speed of the insect? (b) Will the bat be able to catch the insect? Explain.

(a)

To determine
The speed of the insect.

Explanation

Given Info: Bat is flying at 5.00 m/s towards insect. Insect is flying away from the Bat. Bat emits 40.0 kHz chirp and receives 40.4 kHz.

Formula for the frequency observed when the source moves towards the observer is,

fo=fs(v+vovvs)

• fo is the frequency of sound observed by the observer,
• fs is the frequency of the sound emitted by the source,
• vo is the velocity of the observer,
• vs is the velocity of the source,
• v is the speed of sound in air,

Formula for the frequency observed when the bat (source) moves towards the bug (observer) is,

fbug=fbat(v+vbugvvbat)

• fbug is the frequency of sound observed by the bug,
• fbat is the frequency of the sound emitted by the bat,
• vo is the velocity of the bug,
• vs is the velocity of the bat,
• v is the speed of sound in air,

Divide the whole scenario in two steps.

1. 1. Bat acts as a source and the bug acts as an observer,
2. 2. Bug acts as a source and the bat acts as an observer,

In First step, the source is moving towards the observer vs=vbat and observer is moving away from the source vo=vbug So,

Formula to calculate the observed frequency by the bug is,

fobs1=fbat(v+(vbug)v(vbat))=fbat(vvbugvvbat)

• fobs1 is the frequency of sound observed by the bug,
• fbat is the frequency of the sound emitted by the bat,
• vbug is the velocity of the bug,
• vbat is the velocity of the bat,

Now, this frequency is returned to the bat.

In second step, the source is moving away from the observer vs=vbug and the observer is moving towards the source vo=vbat

fobs2=fobs1(v+vbatv(vbug))=fobs1(v+vbatv+vbug)

• fobs2 is the frequency of the sound observed by the bat

Substitute fbat(vvbugvvbat) for fobs1 in the above equation to rewrite it

(b)

To determine
Whether the bat can cat the bug.

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