Question

Consider an airplane trip from Chicago, Illinois, to Denver, Colorado. List some path-dependent functions and some state functions for the plane trip.

Short answer

Path-dependent functions for the plane trip include distance traveled, time taken for the trip, fuel consumption, and work done by the airplane's engines. These functions depend on the specific flight path and other factors such as wind conditions and altitude. State functions for the plane trip include initial and final altitude, initial and final atmospheric pressure and temperature, airplane's change in potential energy, and change in kinetic energy. These functions depend only on the initial and final states of the trip and not on the specific flight path.

Step-by-step solution

Path-Dependent Functions

1. Distance traveled: The airplane's actual distance traveled during the trip will depend on the specific flight path it takes. Different flight paths will result in different distances traveled. 2. Time taken for the trip: The actual time taken for the airplane to travel from Chicago to Denver will depend on the flight path, as well as other factors such as wind conditions and air traffic. 3. Fuel consumption: The amount of fuel consumed by the airplane during the trip will depend on the flight path, wind conditions, altitude, and other factors. 4. Work done by the airplane's engines: The work done by the engines during the trip will depend on the specific flight path and other factors such as wind conditions and altitude.

State Functions

1. Initial and final altitude: These are state functions because they depend only on the initial (Chicago) and final (Denver) locations of the trip, and not on the specific flight path. 2. Initial and final atmospheric pressure and temperature: These values depend only on the initial and final locations of the trip, as well as the altitude, and not the specific flight path. 3. Airplane's change in potential energy: The change in potential energy from the start to the end of the trip only involves the initial and final altitudes of the airplane, making it a state function. \( \Delta PE = m g\Delta h \), where \( \Delta PE \) is the change in potential energy, \(m\) is the mass of the airplane, \(g\) is the gravitational acceleration, and \( \Delta h \) is the change in altitude. 4. Change in kinetic energy: The change in the airplane's kinetic energy depends only on the difference between its initial and final velocities. \( \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2) \), where \( \Delta KE\) is the change in kinetic energy, \(m\) is the mass of the airplane, \(v_i\) is the initial velocity, and \(v_f\) is the final velocity.