Question

A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boat's crew blows a loud horn. By the time the plane's sound detector receives the horn's sound, the plane has travelled a distance equal to one half of its altitude above the ocean. If it takes sound$2s$ to reach the plane, then determine (a) speed of plane, (b) its altitude. Speed of sound as $t=2s$$340m/s$.

Answer 1

Step 1: Analysing given information using diagrammatic representation :

The figure below shows the two positions of the plane, first when the horn was blown and second when the sound was heard. Distance between them is $\frac{h}{2}$and the altitude at which the plane is flying above sea level is $h$. Let the time taken by the sound of the horn to reach the plane is $t$. It is given that $t=2s$ . We need to find $h$ and speed of the plane i.e. $v$ .

Step 2: Formulate equations from and extracted information
It is clear from observation that the time required to reach the horn sound to the plane is equal to the time taken by sound to cover the distance of $\frac{h}{2}$, because only then the plane would have been able to detect it. Using, $speed=\frac{distance}{time}$ we can write

$\frac{h}{2t}=v\Rightarrow h=4v$

Step 3: Substitute values to calculate distance and velocity:

We are given the velocity of sound to be $340m/s$ and the time required to reach the plane to be $2s$.

So, the distance travelled by the sound $d=340\times 2=680\text{m}$.

Looking at the diagram, we can apply the Pythagoras theorem to find $h$.

$\sqrt{h^2+\frac{h^2}{4}}=680$.
Solving this equation for $h$, we get

$h=608.2m$.

From , we obtained $h=4v$, so $v=152.04m/s$.

In the given situation the altitude of the plane will be $608.2m$ above sea level and the velocity of the plane will be $152.04m/s$.