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The supersonic airliner Concorde is 62.1 m long when sittingon
the ground on a typical(15degreesC) day. It is primarilymade of
aluminum. In flight at twice the speed of sound,friction with the
air warms the Concorde's skin and causes theaircraft to lengthen by
25cm. (The passenger cabin is onrollers, and the airplane expands
around the passengers.) What is the temperature of the Concorde's
skin in flight?

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Answer:

[tex]182.74020\ ^{\circ}C[/tex]

Explanation:

L = Initial length = 62.1 m

[tex]\alpha[/tex] = Coefficient of linear expansion of Aluminium = [tex]24\times 10^{-6}/^{\circ}[/tex]

[tex]T_i[/tex] = Initial temperature = 15°C

[tex]T_f[/tex] = Final temperature

[tex]\Delta L[/tex] = Change in length = 25 cm

Change in length due to heat is given by

[tex]\Delta L=L\alpha \Delta T\\\Rightarrow\Delta L=L\alpha (T_f-T_i)\\\Rightarrow T_f=\dfrac{\Delta L}{L\alpha}+T_i\\\Rightarrow T_f=\dfrac{0.25}{62.1\times 24\times 10^{-6}}+15\\\Rightarrow T_f=182.74020\ ^{\circ}C[/tex]

The temperature of the Concorde's skin in flight is [tex]182.74020\ ^{\circ}C[/tex]

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