A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle 60° with ground level. But he finds the aeroplane right vertically above his position. If is the speed of sound, then speed of the plane is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
The sound the person hears was emitted when the plane was to the north at 60° above the ground. By the time that sound reaches the observer, the plane has moved to a position directly overhead.
Let the altitude be h. When the sound was emitted, the horizontal distance from the observer to the plane's position was h/tan60° = h/√3. The sound traveled the slant distance h/sin60° = 2h/√3 at speed v, taking time t = 2h/(√3·v).
In this same time t, the plane traveled the horizontal distance h/√3 at speed u:
u = (h/√3) / (2h/√3·v) = (h/√3) · (√3·v/2h) = v/2.
So the speed of the plane is v/2.
Let the altitude be h. When the sound was emitted, the horizontal distance from the observer to the plane's position was h/tan60° = h/√3. The sound traveled the slant distance h/sin60° = 2h/√3 at speed v, taking time t = 2h/(√3·v).
In this same time t, the plane traveled the horizontal distance h/√3 at speed u:
u = (h/√3) / (2h/√3·v) = (h/√3) · (√3·v/2h) = v/2.
So the speed of the plane is v/2.
[JEE Main 2019 · 12 Jan · Shift I]