Watching a jet plane fly high over my head, I noticed that the sound seemed to be coming from directly above me when the plane was at an angle of 30 degrees from vertical. How fast was the plane flying? Assume the speed of sound was 330 m/s.
Let the plane be H m above the observer.
Time taken for sound to reach observer, t
= H/330 s
Distance plane travelled in time t, D
=H sin(30°) m
Speed of plane
= distance / t
= D/t
= H sin(30°) / (H/330) m/s
= 330 sin(30°) m/s
Convert to km/h if necessary (multiply by 3.6).
To find the speed of the plane, we need to use the concept of the speed of sound and the angle at which the sound appears to be coming from directly above. Let's break down the problem and determine the solution step by step:
Step 1: Understand the problem.
We are given that the sound from the jet plane appeared to be coming from directly above when the plane was at an angle of 30 degrees from vertical. We also know that the speed of sound is 330 m/s. Our goal is to calculate the speed of the plane.
Step 2: Draw a diagram.
Draw a diagram to visualize the situation. Label the angle of 30 degrees, the vertical line, and the line representing the path of the plane.
|\
| \
| \
| \
| \ Plane's
| \ path
| \
| \
| \
(30°) | \
-----------|-------Ground
Step 3: Establish the relationship.
The speed of sound is constant. When the sound reaches our ears, it appears to come from the direction the plane was at that moment. So, the speed of the sound relative to us (on the ground) is also equal to the speed of the plane relative to the ground.
Step 4: Apply trigonometry.
Using trigonometry, we can determine the component of the plane's speed in the vertical direction. We can then relate this component to the speed of sound.
tan(30°) = (component of the plane's speed in the vertical direction) / (speed of sound)
Simplifying this equation, we have:
tan(30°) = (plane's vertical speed) / (330 m/s)
Now we can solve for the plane's vertical speed.
Step 5: Calculate the plane's vertical speed.
Rearranging the equation, we have:
(plane's vertical speed) = tan(30°) * (330 m/s)
Using a calculator, we find that tan(30°) is approximately 0.5774.
(plane's vertical speed) = 0.5774 * (330 m/s)
(plane's vertical speed) ≈ 190.51 m/s
Step 6: Determine the plane's total speed.
Since the plane's vertical speed is just a component of its total speed, we can use trigonometry again to find the total speed.
In a right-angled triangle, the hypotenuse is the square root of the sum of the squares of the other two sides. In this case, the hypotenuse represents the plane's total speed, and the vertical side represents the plane's vertical speed.
So, the plane's total speed is:
(plane's total speed) = √[(plane's vertical speed)^2 + (plane's horizontal speed)^2]
Since the question only asks for the plane's speed, we don't need to consider the horizontal component, and we can safely assume that it is a constant speed without affecting our calculation.
Therefore, the plane's speed is:
(plane's total speed) ≈ (plane's vertical speed) ≈ 190.51 m/s
So, the plane was flying at a speed of approximately 190.51 meters per second.