Treebeard's Stumper Answer
There and Back Again
Dunn Middle School flies east to Washington D.C. on Sunday, yahoo! The Earth is also moving, turning on its axis once a day from west to east at more than 1000 miles per hour on the equator! Will the speed of the Earth's rotation make a difference on our flight times to D.C. and back? We'll learn about time zones firsthand. Another effect of the rotating Earth is the fairly steady jet stream winds that also blow from west to east at our latitude. These winds will help and hinder our flights there and back. Will that make a difference in our total flight time?
|The raw data for this stumper is printed on our airplane tickets. There's a 3 hour time zone difference.|
|4 hours 45 minutes|
5 hours 22 minutes|
Steady winds in one direction will always slow down a trip that goes there and back. Suppose our airplane speed is 500 mph in still air, the wind is blowing there at 100 mph, and the trip distance is 2400 miles each way. We fly there at 600 mph for a time of 4 hours, and our return speed is 400 mph for a time of 6 hours. The total flight time is 4 + 6 = 10 hours. With no wind, our time is 4800 / 500 = 9.6 hours, a bit less. I believe that airplanes also get a small boost traveling with the spin of the Earth from the Coriolis effect. The details are below.
The world would seem a simpler place if we always stayed home. Travel forces us to look closer at our assumptions. Look at any globe. "Up" can't simply be that way (pointing), since the Olympic high jumpers in Sydney, Australia were jumping up in nearly the opposite direction! It's not surprising that Galileo came after Columbus.
Here are our flight paths there and back as near as I could determine looking out the window with my map:
(Base map from MSN Encarta.)
I got our real flight data from my watch and our pilots.
Distance Velocity Cruising Altitude Flight Time LA --> DC:
2040 nautical miles 503 knots maximum
470 knots average
37,000 - 41,000 feet 4 hours, 21 minutes DC --> LA:
2008 nautical miles 470 knots maximum
400 knots average
(120 knots nose-wind)
26,000 - 28,000 feet 5 hours, 10 minutes (Add 15% to get statute miles and miles per hour.)
It's striking that we flew there on a longer flight path and higher, and still beat our time back by 50 minutes. Of course the prevailing west to east winds explain most of the difference. The Earth is turning from west to east, but so is everything else on the Earth, including our airplane. Our flight speed is in addition to the speed of the Earth and wind. At least, that's the simple answer.
Steady winds in one direction will always slow down a round-trip there and back again. I made up the numbers in my answer to make it easy to calculate, but you'll get the same result with any numbers, or a bit of algebra:
t = d/v, the basic equation relating distance, velocity, and time.
s = plane's speed
w = wind speed
d = distance in one direction
d/s time to go there or back with no wind d/s + d/s round trip time with no wind d/(s+w) time to go there with the wind d/(s-w) time to return against the wind d/(s+w) + d/(s-w) round trip time with wind d/(s+w) + d/(s-w) s2 ----------------- = ----- d/s + d/s s2-w2
ratio of flying with wind to flying with no wind
The ratio of round trip times with and without wind is s2/(s2-w2). If the bottom of a fraction is smaller, then the value must be larger. So the round trip time with wind is always greater than without wind. Steady winds both help and hinder, but they hinder for longer, which makes the difference. Don't forget the effects of time in these problems!
There are steady west to east winds at our latitudes because the Earth turns, and those winds make the big difference in our flight times east and west. I believe the Earth's rotation also plays a small part in our flight times east and west because of the Coriolis effect, named for Gaspard Gustave de Coriolis (1792 - 1843).
The Earth turns on its axis once a day. The circumference of the Earth is a bit over 24,000 miles, so the equator is moving at 1000+ miles per hour! We don't ordinarily notice this huge motion because everthing is moving, just like I don't worry about pouring coffee in my moving car, except on the turns. But the Earth is always turning, and it does make a difference when we leave home and travel over large distances. That difference is the Coriolis effect. It effects ocean and air currents as well as airplanes. It's an effect of our rotating frame of reference, but it's a real effect for us.
Coriolis forces are easiest to understand when we move due north and south between latitudes. Different points on the rotating Earth all turn at the same angular speed of 360° per 24 hours, but they move at different tangential speeds, the speed at which an object would be flung into space if the Earth were suddenly to stop in its tracks. That speed depends on your latitude. The Earth spins fastest at the equator because there is more distance to cover in a single rotation. The Earth turns at 1000+ miles per hour at the equator, but it turns a bit less here in Santa Barbara at 34 degrees latitude, only about 1000 x cosine(34°) = 800+ miles per hour, and even less in Washington D.C. at 39 degrees latitude. The poles have no tangental velocity.
An object moving north from the equator keeps its original higher tangential speed, so it has too much rotational speed as it moves further north. The result is that it is carried to the east ahead of its target, as though pushed to the right by a outside Coriolis force. Since Washington D.C. is north of Santa Barbara, this was also a factor in our flight times. Our trip there gets a free boost to the right (east) from the rotation of the Earth. But flying south would give us a similar free boost to the right (now west) since that part of the Earth is now traveling faster than we are, so I think this is symmetrical.
There are Coriolis effects in every direction on the rotating Earth. Drop an egg from a stationary (relative to us on the ground) balloon 1000 feet above. The balloon is keeping up with the turning Earth at a higher elevation and further from the center, so it must be traveling faster in miles per hour. The egg will keep that speed as it falls, so it should land ahead (east) of the point directly below. Another Coriolis effect is that objects moving east experience a force lifting them up, and objects moving west are pulled down. A demonstration of this is to spin a ball on an elastic cord. If you spin faster, the ball goes outward; spin slower and it drops inward. Airplanes experience the same forces. One source on the Web explains:So next time someone says that prevailing east-blowing winds are why east-bound airplane flights are faster than west-bound flights, mention that there's over a 1% difference in effective airplane weight between the two cases, and that might have something to do with it, too!Is this why NASA always launches to the east from Kennedy Space Center in Florida, while our local Vandenberg Air Force Base is reserved for launching payloads into north-south polar orbits? Vandenberg has a clear path to the south because of the east-west trend of the coastline here in Santa Barbara County, California. Years ago, I got to tour the SLC-6 launch complex at Vandenberg, which was to be the launch site for polar-orbiting missions of the Space Shuttle. One day at school we saw the Space Shuttle mockup (the Enterprise?) fly overhead piggyback on a modified 747! The Challenger accident changed everything.
I had trouble with angular momentum and Coriolis forces in college physics, and I still do! Here are a few links for further research to get it right:
- This stumper about how the rotation of the Earth effects flight times has been discussed on the Web, e.g. at PhysLink, The Chinese University of Hong Kong, and MadSci Net (and here). These sources do not all agree.
- There are several sites that discuss Coriolis forces in general terms: the Bad Coriolis page and the Bad Coriolis FAQ (part of Alistair B. Fraser's Bad Science site), Dave Van Domelen's The Coriolis Effect, and an essay from Inwit Publications on The Coriolus Force. The last site has a good general discussion of "fictitious forces" and frames of reference. There are nice animations here and here, and a Java applet to play with. Coriolis forces probably do not effect household toilets and drains, despite The Simpsons.
- I've searched the Web for an explanation of why it's easier for NASA to launch to the east, with the Earth's spin. One NASA document explains that "A launch to the east uses the Earth's rotation somewhat as a springboard." Another NASA source explains that this is a factor in the exact flight inclination from Florida that is chosen for different shuttle missions:Launching close to the equator has the advantage that the velocity the Shuttle picks up from the Earth rotation is maximized. Thus, the 28.4 degree orbit is chosen for Shuttle missions with heavy payloads and missions which do not require a higher inclination.I'm confused. I'll save this for a future stumper!
- I discuss the motion of the Earth in my A Slow Sort of World (11 Feb 00) stumper. There are more tricky distance, velocity, and time stumpers in my Piece of Cake (5 Nov 99), Up This Hill and Down (22 January 1999), and Two DVT Quickies (24 Jan 96).
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