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 traveled a distance equal to one-fourth its altitude above the ocean. Assuming it takes the sound 2.02 s to reach the plane, and taking the speed of sound to be 343 m/s, determine the following.(a) The speed of the plane. (Take the speed of sound to be 343 m/s.) ___m/s (b) What is its altitude? ___m

Answer 1

a The speed of the plane is
`v = 80.851 \ m/s`

b The altitude of the plane is
`Z = 653.27 \ m`

Step-by-step explanation:

Let the altitude of the plane is Z

The distance at which the plane hears the horn is
`d = (1)/(4) * Z`

The time taken for the sound to reach the plane is
`t = 2.02 \ s`

The speed of sound is
`v_s = 343 \ m/s`

The distance from the point the plane hears the sound to the boat is mathematically evaluated as

`D = v_s * t`

substituting values

`D = 2.02 *343`

`D = 692.9 \ m`

This is the diagonal distance of the boat to the plane (hypotenuse)

The altitude of the plane is the vertical distance of the plane from the boat

and the horizontal distance of the plane from the boat is d

Applying Pythagoras theorem

`D^2 = Z^2 + d^2`

=>
`D^2 = Z^2 +[ (1)/(4) * Z ]^2`

substituting values

`692.9^2 = 1.125 Z^2`

=>
`Z^2 =426764.81`

=>
`Z = 653.27 \ m`

=>
`d = (1)/(4) * 653.3`

=>
`d = 163.3 \ m`

Now the velocity of the plane is

`v = (163.3)/(2.02)`

`v = 80.851 \ m/s`