Question

The speed of sound is 340 m/s. A plane flys horizontally at an altitude of 10, 000 m and a speed of 400 m/s. When an observer on the ground hears the sonic boom the horizontal distance from the point on its path directly above the observer to the plane is

Answer 1

6197 m

Step-by-step explanation:

Given a plane is flying at 10,000 m altitude at a speed of 400 m/s, and the speed of sound is 340 m/s, you want to know the horizontal distance from the point directly overhead to the plane when an observer on the ground hears the sonic boom.

Shock wave

The shock wave that creates the sonic boom is the result of constructive interference of the sounds generated by the plane as it flies. The angle α that wave makes with the ground satisfies the relation ...

sin(α) = 1/M

where M is the mach number, the ratio of plane speed to the speed of sound.

For the given speeds, we have ...

sin(α) = 1/(400/340) = 17/20

α = arcsin(17/20) ≈ 58.2°

Distance

When the shock wave passes the observer, the angle of elevation to the plane is α. This means the relationship between the altitude (a) and horizontal distance (d) of the plane is ...

Tan = Opposite/Adjacent

tan(α) = a/d

For an altitude of 10,000 m, the horizontal distance to the plane is ...

d = a/tan(α) = (10000 m)/tan(arcsin(17/20))

d = 10000√111/17 m ≈ 6197 m

The horizontal distance to the plane is 6197 m when the observer hears the sonic boom.