Question

A plane flies at Mach 1.30 , and its shock wave reaches a man on the ground \(3.14 \mathrm{~s}\) after the plane passes directly overhead. Assume that the speed of sound is \(343.0 \mathrm{~m} / \mathrm{s}\) a) What is the Mach angle? b) What is the altitude of the plane?

Short answer

Question: Calculate the altitude of the plane with a Mach number of 1.3 that took 3.14 seconds for its shock wave to reach a person on the ground. Answer: The altitude of the plane is approximately 682.14 meters.

Step-by-step solution

Calculate the Mach angle

The Mach angle is the angle formed between the shock wave and the horizontal surface. Using the Mach number, we can calculate the Mach angle (μ) with the formula: \(\mu = \arcsin \frac{1}{M}\) where \(M\) is the Mach number. In the given exercise, the Mach number \(M = 1.3\). Plugging in the value to the formula, we find the Mach angle: \(\mu = \arcsin \frac{1}{1.3} = \arcsin 0.76923 = 49.44°\) Thus, the Mach angle is 49.44°.

Calculate the distance traveled by the sound wave

We can calculate the distance traveled by the sound wave using the speed of sound and the given time it took to reach the man. The speed of sound, \(v_s = 343.0 ~m/s\), and time taken, \(t = 3.14 ~s\). Distance \(d = v_s × t = 343.0 ~m/s × 3.14 ~s = 1078.62~m\) Therefore, the distance traveled by the sound wave is 1078.62 meters.

Calculate the altitude of the plane

Now, we use the distance traveled by the sound wave and the Mach angle to calculate the altitude of the plane. We can form a right-angled triangle with the Mach angle joining the horizontal ground and the line of the shock wave. The altitude of the plane corresponds to the height (h) of this triangle. As we know, \(\tan \mu = \frac{h}{x}\) We can rearrange the equation to find the altitude (h): \(h = x \times \tan \mu\) Now, we already know the distance traveled by the sound wave (d) and the Mach angle (μ). The distance traveled by the sound wave (d) forms the hypotenuse of the triangle. We can find the base (x) of the triangle using the cosine of the Mach angle (μ): \(x = d \times \cos \mu = 1078.62~m \times \cos 49.44° = 732.81~m\) Now, we have the base (x) and the Mach angle (μ), so we can find the altitude (h): \(h = x \times \tan \mu = 732.81~m \times \tan 49.44° = 682.14~m\) Hence, the altitude of the plane is approximately 682.14 meters.