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Chapter 13: Problem 101

The SR- 71 Blackbird reconnaissance aircraft is primarily made of titanium and typically flies at speeds above Mach \(3 .\) In flight, the length of the SR- 71 increases by about \(0.20 \mathrm{m}\) from its takeoff length of $32.70 \mathrm{m} .$ The average coefficient of linear expansion for titanium over the temperature range experienced by the \(\mathrm{SR}-71\) is $10.1 \times 10^{-6} \mathrm{K}^{-1} .$ What is the approximate temperature of the SR-71 while it is in flight if it started at \(20^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
Answer: The approximate temperature of the SR-71 Blackbird while it is in flight is \(634.08^\circ C\).

Step by step solution

01

Write down the given information

We are given the following information: - Initial length, \(L_0 = 32.70 m\) - Change in length, \(\Delta L = 0.20 m\) - Coefficient of linear expansion, \(\alpha = 10.1 \times 10^{-6} K^{-1}\) - Initial temperature, \(T_0 = 20^\circ C\)
02

Identify the linear expansion formula

The linear expansion formula relates the change in length, initial length, coefficient of linear expansion, and the change in temperature as follows: $$\Delta L = \alpha L_0 \Delta T$$
03

Solve for the change in temperature

We are given all the values except for the change in temperature, so we will isolate \(\Delta T\) in the linear expansion formula: $$\Delta T = \frac{\Delta L}{\alpha L_0}$$
04

Substitute the given values and find the change in temperature

Now we can plug in the given values to find the change in temperature: $$\Delta T = \frac{0.20}{(10.1 \times 10^{-6})(32.70)} \approx 614.08^\circ C$$
05

Calculate the final temperature

We can now find the final temperature, \(T_f\), by adding the change in temperature to the initial temperature: $$T_f = T_0 + \Delta T$$ $$T_f = 20^\circ C + 614.08^\circ C \approx 634.08^\circ C$$ The approximate temperature of the SR-71 Blackbird while it is in flight is \(634.08^\circ C\).

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