Friday, February 15, 2013

"Sir? Can I Just Hover and Let the Earth Turn Towards me?"

There is a prevalent misconception among introductory physics students (not to mention the general public) about flight within our atmosphere.  Admittedly, as a science student long ago, I wondered if I could travel half way around the world along a given latitude in 12 hours simply by hovering in place, say, in a helicopter.  After all, the Earth spins about the axis that extends from its geometric north to south pole once every 24 hours.  I was stunned that it could take more than 12 hours to fly somewhere on the same latitude by airplane.  Were the pilots dummies?  Just sit there and let your destination come to you!

Before addressing this misconception directly, let us investigate just how fast the ground on which you are currently stationed moves with respect to the rotational axis of the Earth.  Using simple kinematics, we can find this relative velocity at the equator, making use of the fact that the radius of the planet is 6,378 km.  The planet rotates through one circumference (40,074 km) in 24 hours, so the relative speed of the surface along the equator is (40,074 / 24 km/h) about 1,670 km/h, or, about twice as fast as a typical commercial airplane cruises.

But, most of us do not live directly along the equator.  We are some angular displacement (latitude) away from it.  If you want to determine the particular surface speed where you reside, multiply 1,670 km/h by the cosine of your particular latitude.  If you are standing in New York City, which is at a latitude of 40.7 degrees, then the land beneath you moves at 1,266 km/h (352 m/s) relative to the spin axis, in a direction perpendicular to it (Eastward).

Can you sense these high speeds?  No.  Organisms can only discern accelerations internally.  If the spin rate of the Earth were to change suddenly, all buildings would fall, oceans would displace, and it would be a really bad day for anyone not living on either geometric pole of the planet.  But, fret not, such an occurrence is extremely unlikely (for an analysis about the amount of energy required to cause the Earth to experience a large angular acceleration, here is a link to a different article).

There are two realities we must accept in order to address the "Why can't I just hover to get around?" question.  The first reality is that a person standing in Central Park has the same relative velocity as the surface on which he or she stands, and, this velocity does not magically vanish when he or she jumps.

Not convinced?  Imagine if this horizontal velocity did disappear the moment we left the surface.  What would you and I see every time a Knicks player jumped straight up inside Madison Square Gardens?  A quick free fall analysis reveals that a vertical jump of 80 cm lasts 0.81 seconds from jump to landing.  In 0.81 seconds, the basketball court beneath the player displaces East by 285 m based on the surface speed in New York City that we calculated earlier.  Fortunately for the basketball player (and the fans in attendance who would prefer that a sweaty seven footer not land on their lap), he also undergoes this same eastward displacement and lands exactly where he took off 0.81 seconds ago, as he also possessed the 352 m/s horizontal component of velocity.

We thus conclude that the Earth`s rotation has no consequence on our horizontal displacement when we jump, because we rotate along with this frame.  Now, let us confront the second and less obvious reality: the Earth`s atmosphere rotates along with the surface.

Imagine for a moment if this were not true, and the atmosphere were static with respect to the rotational axis of the Earth.  In this case, there would be thousand kilometer westerly winds all the time!  Poke your head outside, and, if it is still attached to your neck when you return inside, we can conclude that indeed, the atmosphere does in fact turn along with the planet.  The wind that we do experience is actually a deviation from the air`s continuous eastward velocity.  If there is no wind, then the relative velocity between the air and the surface is zero.

If you should leave the surface and hover in a helicopter, the vehicle imparts a force onto the air, which in turn imparts an equal and opposite force onto it in a vertical sense.  However, in the East/West direction, there are no forces to consider.  The surface and the atmosphere both move along to the East with the same speed, and the helicopter moves along with it when it hovers.  Hovering means staying above the same point, and that is just what happens.  You can hover patiently in New York, but California will not come to you.

Extending this concept to an airplane, there is no advantage in flying halfway around the Earth one way or the other.  You need to displace with respect to the uniform rotation of the atmosphere whether you travel East or West.  The only reason why one direction might be faster than the other is due to wind velocities, which, we recall, are just deviations from the regular eastward rotation.

There is, however, one scenario where the Earth`s rotation can help or hinder space flight: rockets.  When a body leaves the atmosphere and is destined for an orbit of a certain altitude above the Earth, it requires a certain amount of kinetic energy in order to maintain it.  By leaving the rotating frame that consists of the Earth and its atmosphere, it is helpful to use the initial eastward velocity of the Earth towards achieving a West to East orbit, and this is the common practice.  The hundreds of m/s at launch do help, though they are a drop in the bucket compared to the several km/s that a body requires to stay in orbit.

Our inability to detect velocity in an absolute sense can lead us to some deep seeded misconceptions about the way things work.  However, we should consider ourselves lucky that we cannot sense the velocity due to the Earth's rotation, or that of the Earth around the Sun, or that of our solar system around the Milky Way, or that of the Milky Way within our universe; just imagine the meds we'd need to deal with the ensuing motion sickness.  


Julie Plante said...

Good day! Is it because of the "deviation from normal winds" that it takes less time to go to Europe than to come back?? Nothing to do with the direction of the earth's rotation?

The Engineer said...

Yes, it is due to wind direction, not Earth's rotation. Of course, we need to look past any additional confusion due to differing time zones. Here, we are looking at flight times. And flight times are not influenced by Earth's rotation. If they were, they would be influenced A LOT (no commercial airline can keep up with the Earth's absolute rotation speed near the equator). Finally, note that North/South flight times differ due to winds too (and rotation is obviously not a factor there).

May said...

lol, it was my question too.
actually I can say it won't be my (future)kid's question for long time.