8. An airplane is flying at an altitude of 1500 m at mach 3, when will you hear the aircraft? (Calculate distance)

Here's how to figure out when you'll hear the aircraft:<br /><br />1. **Understand Mach Number:** Mach 3 means the airplane is traveling at three times the speed of sound. The speed of sound varies with temperature and altitude, but a reasonable approximation at this altitude is 330 m/s.<br /><br />2. **Calculate the Airplane's Speed:** Speed = Mach Number * Speed of Sound = 3 * 330 m/s = 990 m/s<br /><br />3. **Calculate the Distance to the Airplane:** We can use the Pythagorean theorem to find the slant range (the direct distance from you to the plane). You are at the vertex of a right triangle, with the altitude of the plane (1500 m) as one leg and the horizontal distance the plane travels as the other leg. We're interested in the hypotenuse (the slant range). Let's call the slant range 'd'.<br /><br /> d² = 1500² + x² (where x is the horizontal distance)<br /><br /> Since we're interested in the time you hear the sound, we need to consider the distance the *sound* travels, which is also 'd'.<br /><br />4. **Calculate the Time for Sound to Travel:** Time = Distance / Speed. In this case, the distance is 'd' and the speed is the speed of sound (330 m/s).<br /><br /> t = d / 330<br /><br />5. **Relate the Distances:** While the sound is traveling the distance 'd' to reach you, the plane is continuing to move horizontally. However, for the purpose of this problem, we can assume that the plane's horizontal movement is negligible compared to the slant range, especially since we're using an approximation for the speed of sound. This simplification allows us to treat 'd' as approximately constant for the duration the sound travels.<br /><br />6. **Solve for Time:** Since we're approximating 'd' as constant, we can use the altitude as a close approximation for the distance the sound travels. This is because the horizontal distance the plane travels while the sound reaches you is relatively small compared to the altitude.<br /><br /> t ≈ 1500 m / 330 m/s <br /> t ≈ 4.55 seconds<br /><br />Therefore, you will hear the aircraft approximately 4.55 seconds after it passes directly overhead.<br /><br />**Important Note:** This solution uses approximations. A more precise calculation would involve calculus to account for the plane's continuous movement while the sound travels. However, the approximation provides a reasonably accurate answer.<br />