Let's determine the distance between Auckland and Melbourne.
Firstly, let's calculate the radius of the equator. Its circumference is 40 000 km but let's assume a mistake was made and it's actually only about 38 000 km. Since c = 2πr, r = c/(2π) = 38000/(2π) = 6048 km.
Now, using the law of sines, we can determine the distance (R) between Auckland and the North Pole (σ = 180° - α - γ = 180° - 30° - 42,7° = 107,3°):
r/sin(γ) = R/sin(σ) -> R = (r*sin(σ))/sin(γ)
And knowing this relation, we can finally determine the distance (d) between Auckland and Melbourne (notice that there's a right-angled triangle).
sin(α/2) = d/(2R) -> d = 2Rsin(α/2) = (2*r*sin(σ)*sin(α/2))/sin(γ) = (2*6048*sin(107,3°)*sin(15°))/sin(42,7°) = 4408 km
Bear in mind that this is less than it's in reality because I've disregarded: elevation difference of Auckland and Melbourne, the fact that Melbourne is further from the North Pole than Auckland, and the actually measured circumference of the equator. Plus, it is the length of a straight line when in fact airplanes have to rise and descend and turn left and right. All of these factors make the length even greater in real.
The mean speed of the commercial airplane is v = d/t = 4408/3,5 = 1260 km/h. The maximal speed would be even greater in order to compensate lower speeds during takeoffs/landings. In reality, the
maximum speed of commercial airplanes is only about 1000 km/h.
Moreover, the speed of sound is 1235 km/h. Dear FES, why don't we observe supersonic (and disintegrating) commercial airplanes?