Question

A meteorite hits the surface of the ocean at a speed of \(8800 \mathrm{~m} / \mathrm{s}\). What are the shock wave angles it produces (a) in the air just before hitting the ocean surface, and (b) in the ocean just after entering? Assume the speed of sound in air and in water is \(343 \mathrm{~m} / \mathrm{s}\) and \(1560 \mathrm{~m} / \mathrm{s}\), respectively.

Short answer

Question: A meteorite is traveling at 8,800 m/s just before hitting the ocean surface. The speed of sound in air is 343 m/s, and the speed of sound in water is 1,560 m/s. calculate the shock wave angles produced by the meteorite (a) in the air just before hitting the ocean surface, and (b) in the ocean just after entering it. Answer: (a) The shock wave angle produced by the meteorite in the air just before hitting the ocean surface is approximately 2.24°. (b) The shock wave angle produced by the meteorite in the ocean just after entering it is approximately 10.25°.

Step-by-step solution

(Step 1: Calculate Mach numbers for air and water)

To calculate the Mach number for both cases, we need to use the formula \(M = v_\text{object} / v_\text{sound}\). In this case, the speed of the meteorite (\(v_\text{object}\)) is provided, and we are also given the speed of sound in air (\(v_\text{sound, air}\)) and in water (\(v_\text{sound, water}\)). Calculating the Mach numbers for each medium: Mach number in air, \(M_\text{air} = \frac{8800}{343} ≈ 25.66\) Mach number in water, \(M_\text{water} = \frac{8800}{1560} ≈ 5.64\)

(Step 2: Calculate the shock wave angles)

To find the shock wave angles, we'll use the Mach angle formula, where the Mach angle θ is given by the equation: \(\theta = \sin^{-1}(\frac{1}{M})\). (a) For the shock wave angle in air, we have: Mach angle in air, \(\theta_\text{air} = \sin^{-1}(1 / M_\text{air}) ≈ \sin^{-1}(1 / 25.66) ≈ 2.24°\) (b) For the shock wave angle in water, we have: Mach angle in water, \(\theta_\text{water} = \sin^{-1}(1 / M_\text{water}) ≈ \sin^{-1}(1 / 5.64) ≈ 10.25°\) So, the shock wave angles produced by the meteorite are approximately: (a) \(2.24°\) in the air just before hitting the ocean surface, and (b) \(10.25°\) in the ocean just after entering it.

Key concepts

Understanding the Mach Number

The Mach number, named after the Austrian physicist Ernst Mach, is a dimensionless quantity representing the ratio of an object's speed to the speed of sound in the surrounding medium. It's a critical parameter in aerodynamics and helps in classifying the flight regimes of aircraft, missiles, and meteorites, like in our exercise.

As objects move through a medium, they can travel at subsonic (Mach number < 1), transonic (around Mach 1), supersonic (Mach number > 1 and < 5) or hypersonic (Mach number > 5) speeds. These different regimes affect the air pressure, temperature, density, and many physical phenomena around the object, such as shock waves. Understanding the Mach number is crucial, not just for calculations, but for interpreting the physical behavior of objects moving at high speeds.

In our exercise, we calculated the Mach number for a meteorite in two different media: air and water. With vastly differing speeds of sound in these mediums, the meteorite's speed relative to the speed of sound also changes significantly. This illustrates the importance of the Mach number in understanding how the same object can produce different effects in different environments. Utilization of Mach numbers allows engineers and scientists to predict those effects and design accordingly. For instance, when our meteorite enters the water, its Mach number decreases, due to the higher speed of sound in water, showcasing a significant change in the behavior of the ensuing shock waves.

Speed of Sound Fundamentals

The speed of sound is not a universal constant and can vary significantly with the medium through which it travels and its state (temperature, pressure etc.). It is the speed at which sound waves propagate in a given material. In general, sound travels faster in solids, followed by liquids, and slowest in gases. This is because sound is essentially vibrations traveling through a medium, and the tighter the particles are packed (as in solids), the faster the vibrations can transfer.

For everyday purposes, when we talk about the speed of sound, it's often the value in air at sea level and at a standard temperature of 20°C, which is approximately 343 meters per second. But as we can see from the exercise, the speed of sound in water is much higher, clocked at 1560 meters per second. This significant difference underpins the exercise in calculating the Mach number and subsequently the shock wave angles.

When a meteorite enters the ocean from the air, the shock wave it produces not only changes in strength but also in its propagation velocity, which is directly influenced by the speed of sound in that medium. Understanding the variability of the speed of sound is critical for applications ranging from sonar technology to acoustic engineering, and certainly in understanding phenomena such as the one described in our exercise.

Demystifying the Mach Angle Formula

The Mach angle is a measure of the angle of a shock wave relative to the direction of the object that created it. This concept comes into play when an object moves through a medium faster than the speed of sound, creating what's known as a Mach cone. The Mach angle formula gives us the half-angle at the vertex of this cone, which is inversely related to the Mach number: subtler angles are associated with higher Mach numbers.

The formula is \theta = \(\sin^{-1}(\frac{1}{M})\) where \( \theta \) is the Mach angle and \( M \) is the Mach number. To understand the Mach angle's significance, consider it as a measure of just how 'pointed' or 'wide' the Mach cone will be, which is fundamental in determining how shock waves will hit and impact surrounding objects.

In the case of our meteorite, particularly when calculating \( \theta_\text{water} \) at 10.25°, we see a much wider angle in water due to the lower Mach number, indicating a broader impact range of the shock waves when compared to the 2.24° in air. Such calculations are not only fascinating in the context of natural events, like meteorites, but are also instrumental in designing vehicles and studying impacts in various environments on Earth and beyond.