Question

It is common experience to hear the sound of a low-flying airplane and lookſat the wrong place in the sky to see the plane. Suppose that a plane is traveling directly at you at a speed of 250 mph and an altitude of 2500 feet, and you hear the sound at what seems to be an angle of inclination of 30°. At what angle a should you actually look in order to see the plane? Consider the speed of sound to be 1100 ft/sec.

Answer 1

The actual angle of inclination would be larger than the wrong angle, this is how we get it,

The airplane is flying at 250 mph, which is

`\begin{gathered} (250*5280)/(3600)= \\ 366.7\text{ft/s} \end{gathered}`

When the airplane is at B, it sends sound to the observer on the ground, this sound travels a distance of x,

let's find x,

`\begin{gathered} x=(2500)/(\sin 30) \\ x=5000ft \end{gathered}`

Lets find d, the horizontal distance of the plane initially,

`\begin{gathered} d=5000\cos 30 \\ d=4330ft \end{gathered}`

For the 5000ft the sound traveled, it took some time, which is,

`t=\frac{dis\tan ce}{\text{speed}}=(5000)/(1100)=4.55\text{seconds}`

The plane is moving at 366.7ft/s, in 4.55 seconds it would be at point A, it would have covered a distance equivalent to d - c , this distance is also equal to the airplane's speed times time(4.55 seconds), we have:

`\begin{gathered} d-c=366.7*4.55 \\ d-c=1666.8ft \end{gathered}`

But d= 4330ft , so:

`\begin{gathered} c=4330-1666.8 \\ c=2663.18ft\text{.} \end{gathered}`

So, our unknown angle can be gotten from trigonometrical relations,

`\begin{gathered} \tan y=(2500)/(2663.18) \\ \tan y=0.9387 \\ y=\tan ^(-1)0.9387 \\ y=43.18^o\approx43^o \end{gathered}`

So, you should look up at an angle of 43 degrees, to spot the airplane.