Ribera

Your plane starts from North pole and moves towards the equator. But, since the earth is moving, your plane will experience a sideward force called Coriolis force whose magnitude is given by 2mvw(m is the plane's mass, v is its velocity and w is earth's angular velocity). This force makes the plane to move in a direction which is against the direction of motion of earth, I.e., towards the west. So the relative velocity of the plane starts to increase right from the pole( although it is negligible at the pole). The relative velocity of plane=linear velocity of plane+w/r. r is the radius

One thing should be clear- if we consider only the plane's reference frame, its velocity, wheresoever it may be, is 1000kmph. But it's RELATIVE speed wrt the earth's reference frame is 2670kmph.

This Coriolis force is the problem. As you know it isn't a real force, but a geometrical effect due to rotation of earth beneath a flying object which has its own inertia. This object is "independent" of the earth and that's why its path relative to the ground is like a circle, due to earth rotation. So this "force" cannot modify the speed or the direction of the flying plane, which keeps is own polar speed in a context of increasing ground speed during his flight to the equator.

Starting off, every horizontal direction is due South. If you pick one and maintain it at 1000 km/hr you would leave the Earth at a tangent. If you followed a uniform curve to the equator at the same speed wrt a non rotating Earth frame, shortest path just above the surface, you would not be maintaining due South wrt the rotating Earth frame we generally consider, but would experience an ever increasing "apparent" vector in the Westward direction. This combined with the 1000km/hr Southern vector would put your speed well above 1000 km/hr. This would be "communicated" to the plane by the Earth's atmosphere, and require an exceptional amount of extra power to maintain against the increased drag

 

Without the atmosphere it would be different. The only power required that would be that to produce a vertical (radial) thrust to balance the weight of the aircraft against gravity as without the atmosphere it would have no drag but no lift.

 

You can use vectors to resolve the speed wrt the ground. 1000km/hr South plus 1670 km/hr West when it reaches the equator.

I don't see from where this "ever increasing "apparent" vector in the Westward direction" could come from.

If you are speaking about the atmosphere drag, a very simple vectorial computing gives us (in 2 dimensions)

angle (with the southward direction) = inv tangent (1670/1000) = 59°

speed = SQRT(1000² + 1670²) = 1886 km/h which would make the plane crash.

Whatever it is, the atmosphere drag is not a solution, because if the plane flies straight ahead to the equator, the atmosphere displacement would mean a lateral wind which very fastly would make it to crash.

Hello everyone.

Even simple situations can be not so easy to deal with. I ask myself a simple question and I haven't found an easy explanation.

Imagine a plane taking off from the north pole, and flying at 1000 km/h to the equator. When it arrives there it turns "right", in direction of the west, that is in the opposite direction of earth rotation (it is still flying at same proper speed of 1000 km/h).

The plane doesn't land during that time.

The very simple question is : what will be its speed relatively to the ground ?

At north pole it will have no angular momentum due to earth rotation, because it is located at the axis of earth sphere. Its speed relatively to the ground will then be 1000 km/h. During the travel to the equator, the ground beneath the plane is going faster and faster (due to earth rotation). So at the equator after turning to the west it will get a total speed of 1000 km/h (due to his propulsors) + a relative rotational earth speed of 1670 km/h = 2670 km/h !

That would make the plane to crash instantly, which is something that doesn't happen.

The main problem I see is that it seems that the earth has no mean to "communicate" its rotational speed to the plane which keeps flying at its original polar speed, that is 1000 km/h.