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Classical Mechanics Level 4

On a nice sunny day, John saw a supersonic fighter plane flying parallel to the ground. As a student of Aeronautical Engineering, he knew off-hand the speed of the plane was 1.25 Mach. He could hear its sonic boom, 12 seconds after the plane flew directly overhead. What is the altitude of the plane in km?

  • Assume speed of sound in air to be 330 m/s 330 \text{ m/s} .


The answer is 6.6.

Swapnil Das by Swapnil Das , 20, India

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1 solution

Rohit Ner
Jan 22, 2016

The half angle α \alpha of the mach cone formed by the jet as depicted above is given by
α = sin ⁡ − 1 ( 1 Mach Number ) = 4 5 \begin{aligned}\alpha&=\sin^{-1}\left(\frac{1}{\text{Mach Number}}\right)\\&=\frac{4}{5}\end{aligned}
It is clear that tan of the angle is the ratio of altitude of the jet to the distance travelled by the jet. tan ⁡ α = h 1.25 × 330 × 12 4 3 = h 4950 h = 4950 × 4 3 = 6600 m = 6.6 k m \begin{aligned}\tan \alpha &=\frac{h}{1.25 \times 330 \times12}\\\frac{4}{3} &=\frac{h}{4950}\\h&=4950 \times \frac{4}{3}\\&=6600m\\&\huge\color{#3D99F6}{=\boxed{6.6 km}}\end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Excellent :)

Swapnil Das - 5 years, 4 months ago

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