Question

A stone is dropped inside a well. The sound produced when the stone hits the bottom of the well is heard 4.3 s later. How deep is the well, in meters? The speed of sound is 338 m/s

Answer 1

Here we have given that,

speed of sound = 338 m/s

Let total time from the point of dropping the stone and hearing the sound is T= t1+t2 = 4.3 s

Where

t1 time taken by stone to hit the bottom of well

t2 time taken by sound to return the ground.

The distance travelled by sound will be given as

height of well =h = velocity of sound* time taken by sound to reach the ground

h= c*t2

h= 338×t2 ……(1)

where c is in m/s, t2 in sec. And h is in m.

Now here we are encountering the case of dropping a stone is case of free fall under gravity.

Let initial velocity u=0

t1= time taken by stone to hit water

Hence

h=ut+1/2(gt^2)

h= 0+ 1/2 ( 9.81* (t1)^2)

Taking g= 9.81 m/s^2

h= 4.905(t1)^2……(2)

Equating 1 and 2

338 t2 = 4.905(t1^2)

But t1= T-t2= 4.3-t2

338t2 = 4.905((4.3-t2)^2)

338t2= 4.905( 18.49–8.6t2+ t2^2)

350t2=90.69345–42.183t2+4.905t2^2

4.905t2^2–380.183t2+90.69345=0

On solving this we will get

We will get t2= 0.23 sec.

Hence h= 338*0.23= 77.74 m

Which is our required answer i.e the height of the well.