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The mach number M of an airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure). The mach number is related to the apex angle θ of the cone by the equation below.
Sin(θ/2)=1/M

(a) Find the angle θ that corresponds to a mach number of 1.
θ= _____ degrees
(b) Find the angle θ that corresponds to a mach number of 5.
θ= _____ degrees
(c) The speed of sound is about 760 miles per hour. Determine the speed of an object having the mach numbers from parts (a) and (b).
Mach Number of 1 ____ mph
Mach Number of 5 ____ mph
(d) Rewrite the equation in terms of θ

_________________________________

Respuesta :

Answer:

a)   θ = π

b)   θ  = 24°

c)   760 miles /hour  ,  3800 miles/hour

d)  sin α / 2√ (cos α + 1 )/2*

Step-by-step explanation:

a)  If 1 is a mach number then  1/M  = 1/1  = 1

sin (θ/2) = 1/M     ⇒   sin (θ/2) = 1      ⇒ arcsin (1)   = π/2

Then    (θ/2)  =   π/2       ⇒   θ = π

b)  If 5 is a mach number then  1/M  = 1/5   =  0.2

sin (θ/2) = 1/5    ⇒   arcsin (0.2)      

θ/2  = 12°       ⇒    θ  = 24°

c)  speed of sound is  760 miles/hour

speed of an object with mach number 1   is 760 miles /hour

speed of an object with mach number 5  is  3800 miles/hour

d) We know:

sin2α =  2 sinα cosα       or      sin α  = 2 sinα/2 cosα/2  

sinα/2  =  sin α / 2*cos α/2     (1)

And  cos 2α  =  2cos² α/2 - 1   then         cosα  = 2* cos ²α/2 - 1

cos ²α/2  = ( cosα + 1 )/2    ⇒   cos α/2  =√ (cos α + 1 )/2

Plugging that value in equation 1 we get

sinα/2  =  sin α / 2√ (cos α + 1 )/2*

MrRoyal

Sound wave is caused by the movement of the airplane when passing through a cone as it moves farther from the sound

  • The angle at mach number 1 is [tex]\mathbf{\theta = 180}[/tex]
  • The angle at mach number 5 is [tex]\mathbf{\theta = 23}[/tex]
  • The speeds at both mach numbers are 760 miles per hour, and 3800 miles per hour.
  • The equation in terms of θ is [tex]\mathbf{sin(\frac{\theta}2) = \frac{\frac{sin \theta}{2}}{\sqrt{\frac{cos \theta + 1}{2}}}}[/tex]

The given parameters are:

[tex]\mathbf{sin(\frac{\theta}{2}) = \frac 1M}[/tex]

(a) The angle at mach number 1

This means that M = 1.

So, we have:

[tex]\mathbf{sin(\frac{\theta}{2}) = \frac 11}[/tex]

Divide

[tex]\mathbf{sin(\frac{\theta}{2}) = 1}[/tex]

Take arc sin of both sides

[tex]\mathbf{\frac{\theta}{2} = sin^{-1}(1)}[/tex]

This gives

[tex]\mathbf{\frac{\theta}{2} = 90}[/tex]

Multiply both sides by 2

[tex]\mathbf{\theta = 180}[/tex]

(b) The angle at mach number 5

This means that M = 5

So, we have:

[tex]\mathbf{sin(\frac{\theta}{2}) = \frac 15}[/tex]

Divide

[tex]\mathbf{sin(\frac{\theta}{2}) = 0.2}[/tex]

Take arc sin of both sides

[tex]\mathbf{\frac{\theta}{2} = sin^{-1}(0.2)}[/tex]

This gives

[tex]\mathbf{\frac{\theta}{2} = 11.5}[/tex]

Multiply both sides by 2

[tex]\mathbf{\theta = 23}[/tex]

(c) The speed of sound at mach numbers (a) and (b)

The speed of sound is given at:

[tex]\mathbf{v = 760mih^{-1}}[/tex]

At mach number 1, the speed is:

[tex]\mathbf{v_1 = 760mih^{-1} \times 1}[/tex]

[tex]\mathbf{v_1 = 760mih^{-1} }[/tex]

At mach number 5, the speed is:

[tex]\mathbf{v_5 = 760mih^{-1} \times 5}[/tex]

[tex]\mathbf{v_5 = 3800mih^{-1} }[/tex]

Hence, the speeds at both mach numbers are 760 miles per hour, and 3800 miles per hour.

(d) The equation in terms of θ

We have:

[tex]\mathbf{\frac{sin \theta}{2} = sin(\frac{\theta}2) cos(\frac{\theta}{2})}[/tex]

and

[tex]\mathbf{\frac{cos \theta}{2} =\sqrt{\frac{cos \theta + 1}{2}}}[/tex]

So, the equation [tex]\mathbf{\frac{sin \theta}{2} = sin(\frac{\theta}2) cos(\frac{\theta}{2})}[/tex] becomes

[tex]\mathbf{\frac{sin \theta}{2} = sin(\frac{\theta}2) \times \sqrt{\frac{cos \theta + 1}{2}}}[/tex]

Make [tex]\mathbf{sin(\frac{\theta}2) }[/tex] the subject

[tex]\mathbf{sin(\frac{\theta}2) = \frac{\frac{sin \theta}{2}}{\sqrt{\frac{cos \theta + 1}{2}}}}[/tex]

Hence, the equation in terms of θ is

[tex]\mathbf{sin(\frac{\theta}2) = \frac{\frac{sin \theta}{2}}{\sqrt{\frac{cos \theta + 1}{2}}}}[/tex]

Read more about sound waves at:

https://brainly.com/question/5714146

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