In a jet airplane, the kinetic energy imparted to the...

In a jet airplane, the kinetic energy imparted to the exhaust gases is wasted as far as propelling the airplane is concerned. The useful power is equal to the product of the force available to propel the airplane and the speed of the airplane. If $v$ is the speed of the airplane and $u$ is the relative speed of the expelled gases, show that the mechanical efficiency of the airplane is $\eta=2 v(u+v) .$ Explain why $\eta=1$ when $u=v$

In a jet airplane, the kinetic energy imparted to the exhaust gases is
wasted as far as propelling the airplane is concerned. The useful
power is equal to the product of the force available to propel the airplane and the speed of the airplane. If $v$ is the speed of the airplane and $u$ is the relative speed of the expelled gases, show that the mechanical efficiency of the airplane is $\eta=2 v(u+v) .$ Explain why $\eta=1$ when $u=v$

Key Concepts

Useful Propulsive Power

The useful power available to propel the airplane is calculated as the product of the thrust (force) and the airplane's speed. This concept distinguishes between energy that actually contributes to the motion of the airplane versus energy that is lost to the exhaust, thus underlining the importance of optimizing the speed relations for maximum efficiency.

Relative Velocity and Optimal Efficiency

The analysis involves understanding how the relative speeds of the airplane and the exhaust gases affect energy partitioning. When the relative speed of the expelled gases (u) equals the airplane's speed (v), the wasted kinetic energy in the exhaust becomes minimized and the system achieves maximum mechanical efficiency. In this optimal case, all the energy is utilized for forward propulsion, resulting in an efficiency of 1.

Mechanical Efficiency in Propulsion

This concept refers to the ratio of the useful work output (propulsion power in moving the airplane) to the total energy expended, including energy wasted in the kinetic energy of the exhaust gases. It is a measure of how effectively the engine converts fuel energy into forward motion, taking into account the unavoidable energy losses from expelled exhaust.

Kinetic Energy of Exhaust Gases

In a jet propulsion system, the exhaust gases are expelled at a speed relative to the aircraft, carrying away kinetic energy that does not contribute to moving the airplane. This kinetic energy, imparted to the gases, represents a loss in terms of propulsive work, and its value depends on both the exhaust speed and the airplane’s speed.

Transcript

Please give Ace some feedback