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Chapter 11: Problem 18

An airplane is flying at a flight (or local) Mach number of 0.70 at $10,000 \mathrm{m}$ in the Standard Atmosphere. Find the ground speed (a) if the air is not moving relative to the ground and (b) if the air is moving at $30 \mathrm{km} / \mathrm{hr}$ in the opposite direction from the airplane.

Short Answer

Expert verified
The ground speed of the airplane is (a) 754.74 km/hr when the air is not moving and (b) 724.74 km/hr when the air is moving at 30 km/hr in the opposite direction.

Step by step solution

01

Understanding Provided Information

To understand the exercise, identify the given parameters. The flight Mach number is 0.70, and the airplane is flying at an altitude of 10,000 m. In part (a), the air is not moving relative to the ground and in part (b), the air is moving at 30 km/hr in the opposite direction as the airplane.
02

Use Standard Atmosphere Properties

Knowing that the airplane is flying at a flight Mach number of 0.70 in the Standard Atmosphere, use the properties of the Standard Atmosphere. At 10,000 m, the speed of sound is about 299.5 m/s.
03

Calculate Ground Speed in Static Air

In the first scenario, where the air is not moving relative to the ground, the ground speed of the airplane is equal to the true airspeed. Calculate the true airspeed by multiplying the Mach number by the speed of sound: \(V_{TA} = M \times a\) where \(V_{TA}\) is the true airspeed, \(M\) is the Mach number, and \(a\) is the speed of sound. Thus, \(V_{TA} = 0.70 \times 299.5 = 209.65\) m/s, which equates to \(754.74\) km/hr when converted.
04

Calculate Ground Speed in Moving Air

In the scenario where the air is moving in the opposite direction, the ground speed will decrease due to headwind. Subtract the speed of the moving air from the calculated true airspeed to find the airplane's ground speed: \(V_G = V_{TA} - V_{wind}\) where \(V_G\) is the ground speed, \(V_{TA}\) is the true airspeed, and \(V_{wind}\) is the wind's speed. Thus, \(V_G = 754.74 - 30 = 724.74\) km/hr.

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Most popular questions from this chapter

The gas entering a rocket nozzle has a stagnation pressure of \(1500 \mathrm{kPa}\) and a stagnation temperature of \(3000^{\circ} \mathrm{C}\). The rocket is traveling in the still Standard Atmosphere at \(30,000 \mathrm{m}\). Find the throat and exit area for a flow rate of \(10 \mathrm{kg} / \mathrm{s}\). Assume \(k=1.35, R=\) \(287.0 \mathrm{N} \cdot \mathrm{m} / \mathrm{kg} \cdot \mathrm{K} .\) The gas is perfectly, expanded to the ambient pressure.

A nozzle for a supersonic wind tunnel is designed to achieve a Mach number of \(3.0,\) with a velocity of \(2000 \mathrm{m} / \mathrm{s},\) and a density of \(1.0 \mathrm{kg} / \mathrm{m}^{3}\) in the test section. Find the temperature and pressure in the test section and the upstream stagnation conditions. The fluid is helium.

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Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is \(1 / \sqrt{k}\).

Supersonic airflow enters an adiabatic, constant area (inside diameter \(=1 \mathrm{ft}\) ) 30 -ft-long pipe with \(\mathrm{Ma}_{1}=3.0 .\) The pipe friction factor is estimated to be \(0.02 .\) What ratio of pipe exit pressure to pipe inlet stagnation pressure would result in a normal shock wave standing at (a) \(x=5\) ft, or \((\mathbf{b}) x=10 \mathrm{ft},\) where \(x\) is the distance downstream from the pipe entrance? Determine also the duct exit Mach number and sketch the temperature-entropy diagram for each situation.
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