Question

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

The velocity of the airplane is approximately 96.1 m/s.

Step-by-step solution

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.

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:\[ q = 5675 \, \text{N/m}^2 \]

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.

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:\[ v \approx 96.1 \, \text{m/s} \]

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.

Key concepts

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.

The Pitot tube works by having one end open to the oncoming air stream, allowing it to measure this total pressure. It is crucial for safe airplane operation, as it provides pilots with real-time data to maintain proper speed and control.

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

This formula allows us to calculate the speed of the airplane by measuring the pressures detected by the Pitot tube, making Bernoulli's equation a key tool in aerodynamics for calculating velocity.

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.

Understanding dynamic pressure is crucial for pilots and engineers in ensuring safe and efficient flight.

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.

This calculated velocity is vital for determining how the aircraft should be maneuvered to maintain altitude and ensure navigational precision during flight.

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.

Understanding standard sea level pressure is necessary for interpreting aerodynamic data and optimizing aircraft performance under varying atmospheric conditions.