Chapter 3: Problem 6
A Pitot tube on an airplane flying at standard sea level reads 1.07× 105 N/m2. What is the velocity of the airplane?
Short Answer
Expert verified
The velocity of the airplane is approximately 96.1 m/s.
Step by step solution
01
Understanding Pitot Tube Reading
The Pitot tube measures the total pressure of air, which consists of static pressure and dynamic pressure. At standard sea level, static pressure is given as 101325 N/m² (or Pa). The reading from the Pitot tube is 107000 N/m².
02
Calculate Dynamic Pressure
Dynamic pressure is the difference between total pressure and static pressure. \[ q = P_{total} - P_{static} = 107000 \, \text{N/m}^2 - 101325 \, \text{N/m}^2 \]Calculating this gives:\[ q = 5675 \, \text{N/m}^2 \]
03
Apply Bernoulli's Equation
Bernoulli's equation for incompressible flow relates dynamic pressure to velocity as follows:\[ q = \frac{1}{2} \rho v^2 \]where \( q \) is dynamic pressure, \( \rho \) is air density (1.225 kg/m³ at sea level), and \( v \) is the velocity of the airplane.
04
Solve for Velocity
Rearrange Bernoulli's equation to solve for velocity:\[ v = \sqrt{\frac{2q}{\rho}} \]Substituting the values:\[ v = \sqrt{\frac{2 \times 5675}{1.225}} \]Calculating this gives:\[ v \approx 96.1 \, \text{m/s} \]
05
Conclude with the Airplane's Velocity
The calculated velocity of the airplane is 96.1 m/s based on the Pitot tube reading at standard sea level conditions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pitot tube
The Pitot tube is an essential tool in aerodynamics for measuring airspeed. It is a simple yet effective instrument that determines the speed of an airplane by capturing the difference between the total pressure and the static pressure.
- Total pressure refers to the combination of static and dynamic pressure surrounding the aircraft.
- Static pressure is the atmospheric pressure at the flight altitude, while dynamic pressure is generated by the airplane's movement through the air.
Bernoulli's equation
Bernoulli's equation is a fundamental principle in fluid dynamics that describes the conservation of energy in a flowing fluid. It states that an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy.In its use with a Pitot tube, Bernoulli's equation helps to relate dynamic pressure to velocity:\[q = \frac{1}{2} \rho v^2\]In this equation:
- \( q \) is the dynamic pressure
- \( \rho \) is the air density, which is consistent at sea level (1.225 kg/m³)
- \( v \) is the velocity of the airplane
dynamic pressure
Dynamic pressure is the difference between the total and static pressures and is significant in determining an airplane's speed. It represents the kinetic energy per unit volume of a fluid particle.Dynamic pressure can be expressed mathematically as:\[q = P_{total} - P_{static}\]The airplane's Pitot tube captures this pressure difference, allowing for the calculation of velocity using Bernoulli's equation.
- Measured in pascals (Pa), dynamic pressure increases with the speed of the airplane.
- It is an essential factor for flight dynamics and indicates how much pressure builds up in front of a moving aircraft.
airplane velocity
Airplane velocity refers to the speed at which an aircraft is traveling through the air. It is an essential parameter for the safe operation of airplanes, informing pilots about the aircraft's speed relative to the surrounding air.Velocity is calculated using the relationship between dynamic pressure and density as demonstrated in Bernoulli's equation:\[v = \sqrt{\frac{2q}{\rho}}\]With this equation:
- The term \( q \) stands for dynamic pressure.
- \( \rho \) represents the air density, particularly at standard sea level.
standard sea level pressure
Standard sea level pressure is a critical baseline value used in aerodynamics and meteorology. It is defined as 101,325 N/m² (or Pa). This value is consistently used as a reference for various calculations in aviation and atmospheric science.
- Standard sea level pressure allows pilots and engineers to standardize and compare readings.
- It is used to correct and calibrate instruments such as altimeters and Pitot tubes.
- This reference point simplifies the calculations by providing a common base level of atmospheric pressure.