On a bright winter's day, a coach notices an echo as he jogs between a high school's playing field and the rear wall of its parking structure. He decides to estimate how far he stands from the reflective wall. He abruptly shouts and determines that the sound pulse returns to his as an echo 0.400 seconds later (averaging several trials for enhanced reliability). The air is a frigid −10°C. Six months later, the coach decides to repeat his experiment on a hot summer day, standing at the same exact spot. This time, the echo takes a slightly shorter 0.368 seconds to return to the coach. (a) What is the temperature (in °C) of the air on the summer day? (b)
How far (in m) does the coach stand from the reflective wall?



Answer :

Answer:

Answer:

(a)

=

37.

5

T=37.5

o

C

(b)

=

70.09

x=70.09 m

Explanation:

Speed of Sound

The sound travels in an open space at a predictable speed which mainly depends on the air temperature. It can be calculated with an approximate formula in terms of the temperature T in degrees Celsius.

=

331.3

+

0.606

v

s

=331.3+0.606T

The speed comes in m/s

If we have determined the speed of sound, it's possible to estimate the distance to a large obstacle that could cause a returning sound (echo). The formula for the distance is

=

x=v

s

t

Where

v

s

is the speed of sound and t is half the time we hear our echo, because we can measure the forth and back times, assumed to be equal.

(b) We know the air is at -10^oC, so the speed of sound is

=

331.3

+

0.606

(

10

)

v

s

=331.3+0.606(−10)

=

325.24

/

v

s

=325.24 m/s

The echo was heard 0.431 seconds later, so t=0.431 / 2= 0.2155 seconds

The distance to the rear wall of the gym is

=

(

325.24

)

(

0.2155

)

x=(325.24)(0.2155)

=

70.09

x=70.09 m

(b) On a summer day, the experiment was repeated and the echo was heard at t = 0.396 seconds. Since we already know the distance to the wall, we can estimate the temperature of the air, by solving the equation for T

=

331.3

+

0.606

v

s

=331.3+0.606T

=

331.3

0.606

T=

0.606

v

s

−331.3

We need to compute the new value of

v

s

, with

=

v

s

=

t

x

We know

=

70.09

,

=

0.396

/

2

=

0.198

x=70.09 m,t=0.396/2=0.198 sec

=

70.09

0.198

v

s

=

0.198

70.09

=

354

/

v

s

=354 m/s

Now we can find T

=

354

331.3

0.606

T=

0.606

354−331.3

=

22.7

0.606

T=

0.606

22.7

=

37.

5

T=37.5

o

C