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Mach number

In this exercise we will determine the Mach Number for the different aircraft in the previous table. To help you get started we have calculated the Mach Number for the Concord with a speed of sound v s o u n d = 340 ms - 1 .

For the Condorde we know the speed and we know that:

Mach Number = v aircraft v sound

For the Concorde this means that

Mach Number = 647 340 = 1 . 9
Aircraft speed at altitude ( km · h - 1 ) speed at altitude ( m · s - 1 ) Mach Number
Concorde 2 330 647 1.9
Gripen 2 410 669
Mirage F1 2 573 715
Mig 27 1 885 524
F 15 2 660 739
F 16 2 414 671

Now calculate the Mach Numbers for the other aircraft in the table.

Mach cone

The shape of the Mach Cone depends on the speed of the aircraft. When the Mach Number is 1 there is no cone but as the aircraft goes faster and faster the angle of the cone gets smaller and smaller.

If we go back to the supersonic picture we can work out what the angle of the cone must be.

We build a triangle between how far the plane has moved and how far a wavefront at right angles to the direction the plane is flying has moved:

An aircraft emits a sound wavefront. The wavefront moves at the speed of sound 340 m · s - 1 and the aircraft moves at Mach 1.5, which is 1 . 5 × 340 = 510 m · s - 1 . The aircraft travels faster than the wavefront. If we let the wavefront travel for a time t then the following diagram will apply:

We know how fast the wavefront and the aircraft are moving so we know the distances that they have traveled:

The angle between the cone that forms and the direction of the plane can be found from the right-angle triangle we have drawn into the figure. We know that sin θ = opposite hypotenuse which in this figure means:

sin θ = opposite hypotenuse sin θ = v s o u n d × t v a i r c r a f t × t sin θ = v s o u n d v a i r c r a f t

In this case we have used sound and aircraft but a more general way of saying this is:

  • aircraft = source
  • sound = wavefront

We often just write the equation as:

sin θ = v s o u n d v a i r c r a f t v a i r c r a f t sin θ = v s o u n d v s o u r c e sin θ = v w a v e f r o n t v s sin θ = v w

From this equation, we can see that the faster the source (aircraft) moves, the smaller the angle of the Mach cone.

Mach cone

In this exercise we will determine the Mach Cone Angle for the different aircraft in the table mentioned above. To help you get started we have calculated the Mach Cone Angle for the Concorde with a speed of sound v s o u n d = 340 m · s - 1 .

For the Condorde we know the speed and we know that:

sin θ = v s o u n d v a i r c r a f t

For the Concorde this means that

sin θ = 340 647 θ = sin - 1 340 647 θ = 31 . 7 o
Aircraft speed at altitude ( km · h - 1 ) speed at altitude ( m · s - 1 ) Mach Cone Angle (degrees)
Concorde 2 330 647 31.7
Gripen 2 410 669
Mirage F1 2 573 715
Mig 27 1 885 524
F 15 2 660 739
F 16 2 414 671

Now calculate the Mach Cone Angles for the other aircraft in the table.

End of chapter exercises

  1. In the diagram below the peaks of wavefronts are shown by black lines and the troughs by grey lines. Mark all the points where constructive interference between two waves is taking place and where destructive interference is taking place. Also note whether the interference results in a peak or a trough.
  2. For a slit of width 1300 nm, calculate the first 3 minima for light of the following wavelengths:
    1. blue at 475 nm
    2. green at 510 nm
    3. yellow at 570 nm
    4. red at 650 nm
  3. For light of wavelength 540 nm, determine what the width of the slit needs to be to have the first minimum at:
    1. 7.76 degrees
    2. 12.47 degrees
    3. 21.1 degrees
  4. For light of wavelength 635 nm, determine what the width of the slit needs to be to have the second minimum at:
    1. 12.22 degrees
    2. 18.51 degrees
    3. 30.53 degrees
  5. If the first minimum is at 8.21 degrees and the second minimum is at 16.6 degrees, what is the wavelength of light and the width of the slit? ( Hint: solve simultaneously.)
  6. Determine the Mach Number, with a speed of sound of 340 m · s - 1 , for the following aircraft speeds:
    1. 640 m · s - 1
    2. 980 m · s - 1
    3. 500 m · s - 1
    4. 450 m · s - 1
    5. 1300 km · h - 1
    6. 1450 km · h - 1
    7. 1760 km · h - 1
  7. If an aircraft has a Mach Number of 3.3 and the speed of sound is 340 m · s - 1 , what is its speed?
  8. Determine the Mach Cone angle, with a speed of sound of 340 m · s - 1 , for the following aircraft speeds:
    1. 640 m · s - 1
    2. 980 m · s - 1
    3. 500 m · s - 1
    4. 450 m · s - 1
    5. 1300 km · h - 1
    6. 1450 km · h - 1
    7. 1760 km · h - 1
  9. Determine the aircraft speed, with a speed of sound of 340 m · s - 1 , for the following Mach Cone Angles:
    1. 58.21 degrees
    2. 49.07 degrees
    3. 45.1 degrees
    4. 39.46 degrees
    5. 31.54 degrees

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Maths test. OpenStax CNX. Feb 09, 2011 Download for free at http://cnx.org/content/col11236/1.2
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