Problem 55 - University Physics with Modern Physics

Question

A jet plane flies overhead at Mach 1.70 and at a constant altitude of 950 \(\mathrm{m}\) . (a) What is the angle \(\alpha\) of the shock-wave cone? (b) How much time after the plane passes directly overhead do you hear the sonic boom? Neglect the variation of the speed of sound with altitude.

Step-by-Step Solution

Answer

(a) \(\alpha \approx 36.2^\circ\); (b) The sonic boom is heard \(\approx 2.03\) seconds later.

Step 1: Understanding Mach Angle

When an airplane flies faster than the speed of sound, it generates a shock wave. The Mach angle (\(\alpha\)) is the angle between the direction of the plane's motion and the edge of the shock wave cone. It is given by the formula \(\sin \alpha = \frac{1}{\text{Mach number}}\).

Step 2: Calculating the Mach Angle

Given that the plane is flying at Mach 1.70, we can calculate the angle \(\alpha\). Substitute the given Mach number into the formula: \(\sin \alpha = \frac{1}{1.70}\). Use a calculator to find \(\alpha\); \(\alpha = \arcsin (\frac{1}{1.70})\).

Step 3: Computing the Time for Sonic Boom Detection

The sonic boom is heard when the shock wave cone reaches the observer on the ground. The horizontal distance the shock wave travels is \(d = 950 \cot \alpha\), where 950 m is the altitude of the plane. The speed of the sound, \(v_s\), is approximately 343 m/s. The time \(t\) it takes for the shock wave to travel this horizontal distance is \(t = \frac{d}{v_s}\).

Step 4: Calculating Horizontal Distance

Calculate \(d = 950 \cot \alpha\) using the previously calculated \(\alpha\). \(\cot \alpha\) is the reciprocal of \(\tan \alpha\), which can be found using \(\tan \alpha = \frac{1}{\cot \alpha}\).

Step 5: Finding the Time for Sonic Boom

Substitute the horizontal distance \(d\) into \(t = \frac{d}{343}\) to find the time delay before the observer hears the sonic boom.

Step 6: Calculation Conclusion

With \(\alpha\) calculated, find \(\cot \alpha\), substitute into the horizontal distance \(d = 950 \cot \alpha\), and finally, compute \(t = \frac{d}{343}\) to find the delay time for hearing the sonic boom.

Key Concepts

shock wave

When an object travels through air at a speed greater than the speed of sound, it creates a distinct phenomenon known as a shock wave. This is essentially a high-pressure wave moving along with the object due to its supersonic speed. A shock wave forms because the air molecules cannot move away quickly enough, causing them to be sharply compressed.
Shock waves feature several interesting characteristics:

• They have an abrupt, almost discontinuous change in pressure, temperature, and density of the air they move through.
• These waves move away from the object and propagate in a conical shape, often referred to as a shock cone.
• The speed and intensity of the shock wave depend on the Mach number, which is the ratio of the object's speed to the speed of sound.

Understanding shock waves is important because they have practical implications, especially in aerospace and aviation engineering, as they affect aircraft design and noise reduction strategies.

Mach angle

The Mach angle is an essential element in understanding shock waves. This angle, denoted by \(\alpha\), forms between the direction of motion of a supersonic object and the shock wave cone that the object generates. It is calculated using the formula:\[ \sin \alpha = \frac{1}{\text{Mach number}} \]Effectively, it determines the width of the shock wave cone.The Mach angle has notable features and implications:

• As the Mach number increases (i.e., the object moves faster compared to the speed of sound), the Mach angle decreases.
• This narrower angle implies that the shock wave becomes more focused and intense.
• Understanding the Mach angle is vital for designing aircraft and missiles, as it influences how shock waves propagate and impact the structural integrity of the vehicle.

The Mach angle is crucial in real-world applications such as predicting where and when a sonic boom may be heard by observers on the ground.

sonic boom

A sonic boom occurs when the conical shock wave created by a supersonic object passes over an observer, usually at a distance. People on the ground perceive it as a loud explosion-like noise. What makes sonic booms interesting and complex includes:

• The intensity of the sonic boom is not constant but varies with different factors such as altitude, aircraft speed, and atmospheric conditions.
• Sonic booms can cause discomfort or even structural damage, which is why managing them is important in areas where frequent supersonic travel occurs.
• They only occur when an object is traveling faster than sound and continues as long as the object remains in supersonic flight.

Solving real-world problems involving sonic booms involves understanding their range and effects on both structures and human perception. As aviation technology evolves, so does the need to mitigate sonic boom impacts, making this an area of ongoing research and technological innovation.