Sonic Boom Problem: Find Mach Number of Aircraft

  • Thread starter spursfan2110
  • Start date
In summary, the Mach number of the aircraft is 2.57 based on the given information about the time it takes for the observer to hear the sonic boom and the sound of the rocket engine. This can be calculated by setting the distances traveled by these sounds equal to the distance between the observer and the aircraft, and using the equation for Mach number.
  • #1
spursfan2110
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Homework Statement



A supersonic aircraft is flying parallel to the
ground. When the aircraft is directly over-
head, an observer sees a rocket fired from the
aircraft. 9.28 s later the observer hears the
sonic boom, followed 3.61 s later by the sound
of the rocket engine.
What is the Mach number of the aircraft?

Homework Equations



Vplane = Machplanevsound

And that's all I really know...

The Attempt at a Solution



I'm standed on this one. The homework was covering soundwaves but he didn't talk about sonic waves for more than 3 minutes in class, and the textbook is really really non-descriptive. Any jumpstarts you guys can give me would be great, but I don't see how you're supposed to get anywhere with just two time variables.

Thanks a lot for the patience!
 
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  • #2


Hi there,

First, let's define some variables:
- Vplane = velocity of the aircraft
- Machplane = Mach number of the aircraft
- vsound = speed of sound

We know that the observer hears the sonic boom 9.28 seconds after seeing the rocket fired from the aircraft. This means that the distance between the observer and the aircraft is equal to the distance traveled by the sonic boom in 9.28 seconds. We can represent this as:

Distance = Velocity * Time

So, the distance between the observer and the aircraft is:
Distance = vsound * 9.28 seconds

Similarly, the sound of the rocket engine is heard 3.61 seconds after the sonic boom. This means that the distance between the observer and the aircraft is equal to the distance traveled by the rocket engine sound in 3.61 seconds. We can represent this as:

Distance = Velocity * Time

So, the distance between the observer and the aircraft is:
Distance = Vplane * 3.61 seconds

Since the observer is stationary on the ground, the distance between the observer and the aircraft must be the same in both cases. Therefore, we can set these two equations equal to each other and solve for Vplane:

vsound * 9.28 seconds = Vplane * 3.61 seconds

Solving for Vplane, we get:
Vplane = (vsound * 9.28 seconds) / 3.61 seconds

Now, we know that Mach number is defined as the ratio of the velocity of an object to the speed of sound in the medium it is traveling through. So, we can write:

Machplane = Vplane / vsound

Substituting the value of Vplane that we calculated above, we get:

Machplane = [(vsound * 9.28 seconds) / 3.61 seconds] / vsound

Simplifying, we get:
Machplane = 2.57

Therefore, the Mach number of the aircraft is 2.57.

I hope this helps. Let me know if you need any further clarification or assistance. Good luck with your homework!
 

1. What is the Sonic Boom Problem?

The Sonic Boom Problem refers to the loud, explosive sound created when an aircraft travels faster than the speed of sound. This phenomenon occurs due to the buildup and release of shockwaves as the aircraft breaks the sound barrier.

2. Why is it important to find the Mach number of an aircraft?

The Mach number of an aircraft is crucial for determining the potential for sonic booms. By knowing the Mach number, scientists and engineers can design aircraft that minimize the effects of sonic booms and reduce noise pollution.

3. How is the Mach number of an aircraft calculated?

The Mach number is calculated by dividing the speed of the aircraft by the speed of sound in the surrounding air. This can be expressed as Mach = Aircraft Speed / Speed of Sound.

4. What factors affect the Mach number of an aircraft?

The Mach number of an aircraft can be affected by various factors such as altitude, temperature, and air density. As the altitude increases, the speed of sound decreases, resulting in a higher Mach number. Similarly, colder temperatures and higher air density can also result in a higher Mach number.

5. How can the Mach number of an aircraft be controlled?

The Mach number of an aircraft can be controlled by adjusting the speed of the aircraft or by changing its shape. By reducing the speed or altering the shape of the aircraft, the effects of sonic booms can be minimized. Additionally, further research and advancements in supersonic flight technology may also help in controlling the Mach number of an aircraft.

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