Question

A moth of length 1.0 cm is flying about 1.0 m from a bat when the bat emits a sound wave at 80.0 kHz. The temperature of air is about 10.0∘C. To sense the presence of the moth using echolocation, the bat must emit a sound with a wavelength equal to or less than the length of the insect. The speed of sound that propagates in an ideal gas is given by v=γRTM−−−−√, where γ is the ratio of heat capacities (γ=1.4 for air), T is the absolute temperature in kelvins (which is equal to the Celsius temperature plus 273.15∘C), M is the molar mass of the gas (for air, the average molar mass is M=28.8×10−3kg/mol), and R is the universal gas constant (R=8.314J⋅mol−1⋅K−1).

Find the wavelength λ of the 80.0-kHz wave emitted by the bat.

Express your answer in millimeters.

How long after the bat emits the wave will it hear the echo from the moth?

Express your answer in milliseconds to two significant figures.

Answer 1

Answer: λ = 4.23mm

t (time taken till bat hears echo) = 5.9ms

Explanation:

The velocity of sound waves at temperature T is given by

v = √yRT/M

Substituting the values we get

v = √(1.4) (8.314 J/mol K )(283 K)/(28.8x10^-3kg/mol)

=338.2 m/s

The wavelength is given by,

λ = v/n

λ = 338.2 m/s/80.0x10^3 Hz

= 4.23 x10^-3m

= 4.23 mm

The time taken is given by,

t=d/v

We will multiply d by factor 2 because of the reflection of waves.

= (2)(1m)/ 338.2 m/s

= 0.00591s

=5.9 ms