Answer
Verified
466.5k+ views
Hint: In the question there is a jet travelling at mach 3.00. a person is standing directly under the jet. The height from the observer to the jet is given to us. We need to find the time taken for the shock wave to reach the observer. We know the equation to find the shock angle and from the figure we can find the change in position of the jet. Using this we can derive for velocity and thus get the time taken.
Formula used:
Shock angle,
$\sin \theta =\dfrac{v}{{{v}_{s}}}$
Velocity,
${{v}_{s}}=\dfrac{dx}{dt}$
Complete answer:
In the question it is said that a supersonic jet is travelling at an altitude ‘h’ and h is given to us.
h = 20000
The mach number of the jet is given to us as 3.00
We know that the average speed of sound in air is 335 m/s.
In the figure the angle ‘$\theta $’ is the shock angle.
We know the equation for the shock angle,
$\sin \theta =\dfrac{v}{{{v}_{s}}}$, where ‘$v$’ is the velocity of the sound in air and ‘${{v}_{s}}$’ is velocity of the mach.300
We know that velocity of mach is the product of Mach number and velocity of sound in air, i.e. ${{v}_{s}}=3\times 335$
Therefore we get,
$\begin{align}
& \sin \theta =\dfrac{335}{3\left( 335 \right)} \\
& \sin \theta =\dfrac{1}{3} \\
\end{align}$
From this we get the shock angle as,
$\begin{align}
& \theta ={{\sin }^{-1}}\left( \dfrac{1}{3} \right) \\
& \theta ={{19.47}^{o}} \\
\end{align}$
From the figure we get tan of the angle ‘$\theta $’ as,
$\tan \theta =\dfrac{h}{x}$, where ‘h’ is the altitude of the jet and ‘x’ is the distance that the jet moves.
We know the value of ‘$\theta $’ and ‘h’, hence we can find ‘x’ as,
$x=\dfrac{h}{\tan \theta }$
$\begin{align}
& x=\dfrac{20000}{\tan \left( 19.47 \right)} \\
& x=5.6572\times {{10}^{4}}m \\
\end{align}$
This is the change in position of the jet.
We need to find the taken by the observer to hear the sound.
We now know that velocity is the derivative of position with respect to time.
Therefore here we have,
${{v}_{s}}=\dfrac{dx}{dt}$
In this above equation we know the values of ${{v}_{s}}$ and ‘x’, hence we get time as,
$dt=\dfrac{5.6572\times {{10}^{4}}}{335\times 3}$
$dt=56.29\sec $
This is the time taken by the observer to hear the shock wave.
Note:
When a wave is propagating and the medium of propagation does not change, then there is no change in the velocity of the sound wave. But if there is a relative motion between the source and the observer, then the frequency of the wave will change accordingly. This phenomenon is called the Doppler Effect in sound.
While solving the question we take velocity as the derivative of position with respect to time. In this case the change in position is the value marked as ‘x’ in the figure.
Formula used:
Shock angle,
$\sin \theta =\dfrac{v}{{{v}_{s}}}$
Velocity,
${{v}_{s}}=\dfrac{dx}{dt}$
Complete answer:
In the question it is said that a supersonic jet is travelling at an altitude ‘h’ and h is given to us.
h = 20000
The mach number of the jet is given to us as 3.00
We know that the average speed of sound in air is 335 m/s.
In the figure the angle ‘$\theta $’ is the shock angle.
We know the equation for the shock angle,
$\sin \theta =\dfrac{v}{{{v}_{s}}}$, where ‘$v$’ is the velocity of the sound in air and ‘${{v}_{s}}$’ is velocity of the mach.300
We know that velocity of mach is the product of Mach number and velocity of sound in air, i.e. ${{v}_{s}}=3\times 335$
Therefore we get,
$\begin{align}
& \sin \theta =\dfrac{335}{3\left( 335 \right)} \\
& \sin \theta =\dfrac{1}{3} \\
\end{align}$
From this we get the shock angle as,
$\begin{align}
& \theta ={{\sin }^{-1}}\left( \dfrac{1}{3} \right) \\
& \theta ={{19.47}^{o}} \\
\end{align}$
From the figure we get tan of the angle ‘$\theta $’ as,
$\tan \theta =\dfrac{h}{x}$, where ‘h’ is the altitude of the jet and ‘x’ is the distance that the jet moves.
We know the value of ‘$\theta $’ and ‘h’, hence we can find ‘x’ as,
$x=\dfrac{h}{\tan \theta }$
$\begin{align}
& x=\dfrac{20000}{\tan \left( 19.47 \right)} \\
& x=5.6572\times {{10}^{4}}m \\
\end{align}$
This is the change in position of the jet.
We need to find the taken by the observer to hear the sound.
We now know that velocity is the derivative of position with respect to time.
Therefore here we have,
${{v}_{s}}=\dfrac{dx}{dt}$
In this above equation we know the values of ${{v}_{s}}$ and ‘x’, hence we get time as,
$dt=\dfrac{5.6572\times {{10}^{4}}}{335\times 3}$
$dt=56.29\sec $
This is the time taken by the observer to hear the shock wave.
Note:
When a wave is propagating and the medium of propagation does not change, then there is no change in the velocity of the sound wave. But if there is a relative motion between the source and the observer, then the frequency of the wave will change accordingly. This phenomenon is called the Doppler Effect in sound.
While solving the question we take velocity as the derivative of position with respect to time. In this case the change in position is the value marked as ‘x’ in the figure.
Recently Updated Pages
How is abiogenesis theory disproved experimentally class 12 biology CBSE
What is Biological Magnification
Explain the Basics of Computer and Number System?
Class 11 Question and Answer - Your Ultimate Solutions Guide
Write the IUPAC name of the given compound class 11 chemistry CBSE
Write the IUPAC name of the given compound class 11 chemistry CBSE
Trending doubts
Proton was discovered by A Thomson B Rutherford C Chadwick class 11 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE
State the laws of reflection of light
What are ekaboron ekaaluminium and ekasilicon class 11 chemistry CBSE
Three moles of B2H6 are completely reacted with methanol class 11 chemistry CBSE