Question

2. (Navigation on a curved Earth): How long will it take to fly from Portland, Oregon (45 N and 120 W) to Tashkent, Uzbekistan (45 N and 60° E) in the shortest possible time? Assume an altitude of 10 km, a speed of 800 km/hr and a spherical Earth. Also, describe the trajectory

Answer 1

In the given problem, we have

Constant latitude 45°. Of both the cities.

But, we have different longitudes of both the cities.

We will find the angular speed of the plane to determine the time of flight.

To find the angular velocity of the plane, we need the radius of the rotation of the plane about the center of the earth, because we are provided with Altitude and the linear velocity of the plane.

Therefore,

Assuming earth to be a sphere,

Since the radius of the earth in km is = 6371 km

Radius of rotation of the flying plane = Radius of earth + Altitude of flight.

= 6371 + 10

= 6381 km

Therefore

Angular velocity of the plane will be,

Linear velocity/Radius of rotation

= 800/6381 radians/hour

= 800*180/(π*6381) degrees/hour.

[since 1 radian = 180/π degrees]

= 7.18329 degrees/hr

The angular distance, that the plane has to cover can be calculated by the difference between the longitudes.

i.e.

120°W to 60°E

Since the longitudes are in opposite directions, the angular distance will be the sum off the values of the longitudes.

= 180°

Time of flight

= angular distance / angular speed

= 180/7.1823 hours

= 25.0613 hours.

Therefore the time of flight will be 25.0613 hours.

The trajectory will be a circular arc, because we are given a constant altitude as well as a constant latitude.

Note that, we have not taken the angular velocity of the earth, as the plane is also rotating with the earth with the same velocity. Therefore the have zero relative angular velocity.