# Applying newton's first law for a plane flying over the earth

From a quick google search:

Newton's first law states that, if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a force.

This is simple enough and let's us know how to detect the presence of force.

Now consider a plane moving above the surface of earth(*), if we look at the path of the plane from space, it is true that the path of the plane is curved and hence there must be some force changing it's trajectory by first law. However, to the pilot in the plane it seems that it is moving straight line as everywhere is flat in the pilot's field of vision and therefore no force is acting on it by first law.

Therefore, what's straight in one perspective is not straight in another perspective. Hence, does newton's law fail when an object moves on a curved surface? If so, how do we fix it?

*: neglect rotation and revolution of earth

The forces of gravity $$F_g$$ and lift $$F_l$$ act on the plane in opposite directions. $$F_g - F_l = F_{cent}$$ where $$F_{cent}$$ is the centripetal force keeping the plane moving in a circular path with respect to the earth. $$F_{cent} = mv^2/r$$ where $$m$$ is the mass of the plane, $$v$$ is its velocity, and $$r$$ is the distance from the center of mass of the plane to the center of mass of the earth. The plane moves at velocity v due to the thrust force from its engines countering the forces of "drag" from the air. [Note. The plane has insufficient velocity to allow it to remain at a stable $$r$$ without any lift; that is; it is not in "orbit" about the earth like the moon.]