Question

A hiker determines the length of a lake by listening for the echo of her shout reflected by a cliff at the far end of the lake. She hears the echo 3.3s after shouting. The speed of sound in air is 343 m/s.

 

Determine the length of the lake.

Express your answer to two significant figures and include the appropriate units.

L =

Answer 1

Concepts and reason

The concepts used to solve this question is the speed of the sound in air.

Initially, determine the distance travelled by the sound by using the relation between speed, distance and time. Later calculate the length of the lake by using the calculated distance of the sound wave.

Fundamentals

The speed of the sound depends on the elasticity, density and the temperature of the medium. At the room conditions, the speed of sound is about $343{\rm{ m/s}}$ . The speed of sound is directly proportional to the temperature of the medium. More, the temperature more will the speed of sound in that medium.

The relation between the speed, time and distance is,

$v = \frac{d}{t}$

Here, $v$ is the speed of the object, $d$ is the distance travelled by the object and $t$ is the time taken by the object to cover the distance.

The distance travelled by the sound is,

$d = vt$ …… (1)

Here, $v$ is the speed of sound and $t$ is the time.

Substitute $343{\rm{ m/s}}$ for $v$ and $3.3{\rm{ s}}$ for $t$ in the equation (1).

$\begin{array}{c}\\d = vt\\\\ = \left( {343{\rm{ m/s}}} \right)\left( {3.3{\rm{ s}}} \right)\\\\ = 1131.9{\rm{ m}}\\\end{array}$

The sound wave travels back and forth across the lake. Hence, the length of the lake is,

$L = \frac{d}{2}$ …… (2)

Substitute $1131.9{\rm{ m}}$ for $d$ in the equation (2).

$\begin{array}{c}\\L = \frac{d}{2}\\\\ = \frac{{1131.9{\rm{ m}}}}{2}\\\\ = 565.95{\rm{ m}}\\\end{array}$

The length of the lake in two significant figures is,

$L = 570{\rm{ m}}$

Ans:

The length of the lake is ${\bf{570 m}}$