A jet aircraft travelling at Mach 3 at a height of 15 km passes directly over an observer on the ground. Calculate the distance between the plane and the observer when they first h... A jet aircraft travelling at Mach 3 at a height of 15 km passes directly over an observer on the ground. Calculate the distance between the plane and the observer when they first hear the sound.
Understand the Problem
The question is asking us to calculate the distance between a jet aircraft flying at Mach 3 at a height of 15 km and an observer on the ground when the observer first hears the sound. This involves understanding the relationship between the speed of sound, the speed of the aircraft, and the geometry of the situation.
Answer
The distance between the plane and the observer when they first hear the sound is approximately $44952.57 \, \text{m}$.
Answer for screen readers
The distance between the plane and the observer when they first hear the sound is approximately $44952.57 , \text{m}$.
Steps to Solve
- Determine the speed of sound
The speed of sound in air at sea level is approximately $343 , \text{m/s}$.
- Calculate the speed of the jet
Since the jet is flying at Mach 3, its speed can be calculated as:
$$ \text{Speed of the jet} = 3 \times 343 , \text{m/s} = 1029 , \text{m/s} $$
- Calculate the time it takes for sound to reach the observer
The sound travels in a straight line from the jet to the observer. The distance the sound must travel is the height of the jet, which is 15 km or 15,000 m. The time taken for sound to travel that distance is:
$$ t = \frac{\text{distance}}{\text{speed of sound}} = \frac{15000 , \text{m}}{343 , \text{m/s}} \approx 43.7 , \text{s} $$
- Calculate the horizontal distance the jet travels during that time
During the time it takes for the sound to reach the observer, the jet is still moving. The horizontal distance traveled by the jet is:
$$ \text{distance}{jet} = \text{speed}{jet} \times t = 1029 , \text{m/s} \times 43.7 , \text{s} \approx 44900 , \text{m} $$
- Use the Pythagorean theorem to find the direct distance
The total distance between the jet and the observer when the observer first hears the sound can be found using the Pythagorean theorem:
$$ \text{distance}_{total} = \sqrt{(15,000 , \text{m})^2 + (44,900 , \text{m})^2} $$
Calculating this gives:
$$ \text{distance}_{total} \approx \sqrt{225000000 + 20161001000} \approx \sqrt{20183201000} \approx 44952.57 , \text{m} $$
The distance between the plane and the observer when they first hear the sound is approximately $44952.57 , \text{m}$.
More Information
This calculation illustrates how sound travels slower than the jet, creating a delay before the observer hears it. The large distance demonstrates the effects of speed and height in aerodynamics.
Tips
- Confusing the speeds of the jet and sound; ensure to use the correct speed values.
- Not converting all values to the same units.
- Misapplying the Pythagorean theorem; ensure the correct sides are used.
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