An aircraft flies from an airport X(35°S, 40°E) and after 1500km due East,it reaches an airport Y. It then fly due south of another airport Z on latitude 70°S. Calculate, correct to 3 s.f;
(a) Radius of the latitude through X
(b) The longitude of Y
(c) The distance between Y and Z
(d) The speed of Z due to the rotation of the Earth
(e) The distance of Z from North
If the earth's radius is R, then
at latitude θ the radius is R cosθ
each degree of latitude is 111 km
Now you should be able to answer the questions.
To find the answers to the given questions, we will use the following formulas and information:
1. The Earth's radius is approximately 6,371 km.
2. The latitude lines are circles that run parallel to the equator, and the radius of each latitude circle is smaller than the Earth's radius. The radius of a latitude circle can be calculated using the formula:
radius = Earth's radius * cosine(latitude)
3. The longitude lines are semicircles that connect the North and South poles. The distance between any two longitudes can be calculated using the formula:
distance = (longitude2 - longitude1) * (Earth's circumference / 360°)
4. The distance between two airports can be calculated using the Pythagorean theorem:
distance = sqrt((distance_latitude)^2 + (distance_longitude)^2)
5. The tangential speed of a point on Earth's surface due to its rotation can be calculated using the formula:
speed = (angular velocity of Earth) * (radius of latitude circle)
Now let's solve each question step-by-step:
(a) Radius of the latitude through X:
radius_X = Earth's radius * cosine(latitude_X)
radius_X = 6371 km * cos(35°)
radius_X ≈ 4,257.77 km
(b) Longitude of Y:
The aircraft flew due East for 1500 km, which corresponds to a change in longitude.
distance_longitude_Y = 1500 km * (360° / (Earth's circumference))
distance_longitude_Y ≈ 0.351°
longitude_Y = longitude_X + distance_longitude_Y
longitude_Y = 40°E + 0.351°
longitude_Y ≈ 40.351°E
(c) Distance between Y and Z:
The aircraft flew due South from Y to Z, which corresponds to a change in latitude.
distance_latitude_YZ = 70°S - 35°S
distance_latitude_YZ = 35°
radius_Y = Earth's radius * cosine(latitude_Y)
radius_Y = 6371 km * cos(70°S)
radius_Y ≈ 981.83 km
distance_YZ = sqrt((distance_latitude_YZ)^2 + (distance_longitude_Y)^2)
distance_YZ = sqrt((35)^2 + (0)^2)
distance_YZ ≈ 35 km
(d) Speed of Z due to Earth's rotation:
The angular velocity of Earth is approximately 0.0000727 rad/s.
speed_Z = (angular velocity of Earth) * (radius of latitude circle at Z)
speed_Z = 0.0000727 rad/s * radius_Y
speed_Z ≈ 0.0712 km/s
(e) Distance of Z from North:
The distance of Z from North is equal to the difference in latitude between Z and the North pole.
distance_North_Z = 90° - 70°S
distance_North_Z = 70°
Thus, the answers are:
(a) The radius of the latitude through X is approximately 4,257.77 km.
(b) The longitude of Y is approximately 40.351°E.
(c) The distance between Y and Z is approximately 35 km.
(d) The speed of Z due to the rotation of the Earth is approximately 0.0712 km/s.
(e) The distance of Z from North is approximately 70°.
To solve these questions, we can use basic trigonometry and the properties of coordinates on the globe. Let's go through each question step by step:
(a) Radius of the latitude through X:
The radius of a latitude is the distance from the center of the Earth to that latitude. Since the coordinates of airport X are given as (35°S, 40°E), it is located 35 degrees south of the Equator. The radius of the Earth at that latitude can be calculated using the formula:
R = R_earth * cos(latitude)
where R_earth is the average radius of the Earth (approximately 6371 kilometers). Plugging in the values:
R = 6371 km * cos(35°)
Calculating this expression will give you the radius of the latitude through X.
(b) The longitude of Y:
Given that the aircraft flew 1500 km due east from airport X, we can calculate the new longitude of Y by adding the distance traveled to the initial longitude of X. In this case, the initial longitude is 40°E. Therefore, the longitude of Y can be found by:
Longitude of Y = 40°E + (1500 km / (circumference of Earth) * 360°)
(c) The distance between Y and Z:
To calculate the distance between two points on a sphere (such as the Earth), you can use the Haversine formula. The formula is given by:
d = 2 * R_earth * arcsin(√(sin²((φ₂-φ₁)/2) + cos(φ₁) * cos(φ₂) * sin²((λ₂-λ₁)/2)))
where φ₁ and φ₂ represent the latitudes of points Y and Z, respectively, and λ₁ and λ₂ represent their longitudes. Plug in the values of coordinates Y and Z to calculate the distance between them.
(d) The speed of Z due to the rotation of the Earth:
The speed of a point on the Earth's surface due to rotation depends on its latitude. The formula to calculate this speed is:
Speed = R_earth * cos(latitude) * angular speed of Earth
where the angular speed of Earth is approximately 0.0041667 degrees per second.
(e) The distance of Z from North:
The distance of Z from the North Pole can be calculated by subtracting the latitude of Z from 90 degrees, since the latitude of the North Pole is 90 degrees.