Which way do I go

[fast]

Let’s confine the parameters to the continental United States. Pick a point towards the middle such that the point can get no further from the border. Then, draw a circle around the point such that it can be no larger without crossing a border. I don’t know what the radius would be in miles, but i’ll just use a nice round number and call it 500 miles. The distance from the most northern point to the most southern point would be 1000 miles. East to west would be 1000 miles. In fact, every point would be equal distance to its opposite.

Now, let’s say the entire area of the circle was paved and flat and we ran a car race where we both averaged 100 MPH. We would essentially tie every time we raced. What’s most important here is that it wouldn’t matter what direction we raced. We always tie.

Now, you decide to take to the skies while I remain below. We run 360 races (one for each degree of the circle). In fact, it’s like running 180 courses twice—one one way and one the other way. No matter what, we always average 100MPH.*

But, we don’t tie anymore with one exception. You beat me each and every time except for the one time we tie. If I want to run the race again (and not lose) where do I start, and where do I finish? Essentially, what direction do I go? Which of the 180 tracks do I choose and which way do I run it?

 

[fast]

*Correction. I had time to edit, but my correction is complicated to explain. Yes, we both averaged 100MPH on land, but I shouldn’t say what I said about the sky. Let me put it another way. The engines have the same power and one has no advantage over the other. The curvature of the earth and its direction of spin is what’s responsible for varying distance traveled and thus speed obtainable.

 

[fast]

The answer I’m looking for is the course that follows the planets spin. I want to drive in the direction that it’s spinning, but I don’t know what direction that is.

 

[steve_bank]

A race is between two point's on a specified course. If you watch the Indy 500 or NASCAR races two cars do not necessarily take the exact same path around the track. Two cars can have the same average speed but with different elapsed times.

The question is simplified, if two velocity functions v1(t) and v2((t) can have the same average velocity but different elapsed times on a straight line course between two points p1 and p2. Like a drag race.

Mathematically the path does not matter. The point by point velocity curve v(t) is integrated to get distance and time. Straight line path, circle, parabola. All the same.

I can't answer that off the top of my head. I could try different functions which would be easy, but It would take the form of a proof to be sure.

 

[fast]

A race is between two point's on a specified course. If you watch the Indy 500 or NASCAR races two cars do not necessarily take the exact same path around the track. Two cars can have the same average speed but with different elapsed times.

The question is simplified, if two velocity functions v1(t) and v2((t) can have the same average velocity but different elapsed times on a straight line course between two points p1 and p2. Like a drag race.

Mathematically the path does not matter. The point by point velocity curve v(t) is integrated to get distance and time. Straight line path, circle, parabola. All the same.

I can't answer that off the top of my head. I could try different functions which would be easy, but It would take the form of a proof to be sure.

If you are flying towards a destination that is coming at you, that is different than flying towards a destination that is going away from you. I would think (all else being equal) that you get to a destination coming at you quicker than a destination going away from you.

Thing is, with flights, we generally aren’t going exactly one way or the other, and that’s because we are usually neither going in the exact direction of Earth’s spin or it’s opposite.

 

[steve_bank]

A race is between two point's on a specified course. If you watch the Indy 500 or NASCAR races two cars do not necessarily take the exact same path around the track. Two cars can have the same average speed but with different elapsed times.

The question is simplified, if two velocity functions v1(t) and v2((t) can have the same average velocity but different elapsed times on a straight line course between two points p1 and p2. Like a drag race.

Mathematically the path does not matter. The point by point velocity curve v(t) is integrated to get distance and time. Straight line path, circle, parabola. All the same.

I can't answer that off the top of my head. I could try different functions which would be easy, but It would take the form of a proof to be sure.

If you are flying towards a destination that is coming at you, that is different than flying towards a destination that is going away from you. I would think (all else being equal) that you get to a destination coming at you quicker than a destination going away from you.

Thing is, with flights, we generally aren’t going exactly one way or the other, and that’s because we are usually neither going in the exact direction of Earth’s spin or it’s opposite.

In differential equations that is called a problem in related rates. Two trains are headed towards each other at different speeds, where do they collide? A car is at a certain speed trying to cross a train track ahead of a train and the train is going a certain speed, does the car make it?

If you are flying with or against the Erath's rotation it is the same problem. You end up with a differential equation that yields ground speed.

Same problem if you are traveling through space to another planet. Spaceship speed + planet speed = intersection point in time. The equation is a little more complicated but not much. Obviously if the ship speed is too slow it can never get to tyeh planet.

A train is going 50mph on a straight track and is 10 miles from a crossing. A car is on a road to the crosing20 miles away. How fast does the car have to go to beat the train?

 

[fast]

If you are flying with or against the Erath's rotation it is the same problem. You end up with a differential equation that yields ground speed.

I’m not sure how that all works. I’m banking on there being a difference.

 

[steve_bank]

If you are flying with or against the Erath's rotation it is the same problem. You end up with a differential equation that yields ground speed.

I’m not sure how that all works. I’m banking on there being a difference.

Air speed is relative to the ground.

 

[fast]

If you are flying with or against the Erath's rotation it is the same problem. You end up with a differential equation that yields ground speed.

I’m not sure how that all works. I’m banking on there being a difference.

Air speed is relative to the ground.

Right, but the ground is moving.

If I fly from one point to another that’s 1000 miles away, and if I choose the shortest route, are there not a couple points where the path is straight and not curved?

 

[fast]

If the planet is rotating west to east, does this mean flying either directly east (or directly west) means the flight path would have no arc?

 

[bilby]

In the absence of atmospheric effects, your momentum at your starting point is such that flying east or west is irrelevant to your time between two points on the Earth's surface. However the wind tends to be pushed around the planet by the sun; So in practice, tailwinds can be anticipated to be more likely when flying from west to east - although that's not always going to be true.

If your objective is to achieve orbital velocity (which is independent of your movement across the ground) you should work with your eastward momentum, and head east from your launch point (this is why spacecraft are most commonly launched from sites on the east coast of a continent, and at low latitudes (ie close to the equator). The closer to the equator you are, the more easterly momentum you start with - and by launching from the east coast, you ensure that the earliest part of your flight is over the ocean, so a failed launch crashes into the water, rather than onto land.

There are exceptions to both of these typical choices for launch sites. If you want your satellite to have an orbit that extends a long way north and south, then a higher lattitude launch (and/or a launch more in the direction of the nearer pole) is better; Spy satellites get coverage of high latitudes in this way, which is why the US military uses Vandenberg, launching in a northwesterly trajectory over the Pacific. But that's more expensive than an easterly launch path from Cape Canaveral, or from the ESA launch sites in French Guiana.

And oceans aren't the only uninhabited areas where you can crash your failed rockets - the Soviets (and now the Russians) launched from Kazakhstan, which has a lot of mostly empty steppes to its east.

 

[fast]

Dang, orbital velocity. I knew I had forgotten something.

 

[steve_bank]

Air speed is relative to the ground.

Right, but the ground is moving.

If I fly from one point to another that’s 1000 miles away, and if I choose the shortest route, are there not a couple points where the path is straight and not curved?

Back to relativity- inertial reference frames and relative velocities. The jet moves relative to theEarth surface. Both jet and Earth revolve around the sun. Jet, sun, and Earth revolve around the galaxy...

You can be the jet as an stationary x,y,z coordinate system and that is still with everything else moving. Jet standing still and Earth moving. Or you can pick the surface as a reference and the jet moves relative to the ground. It doesn't matter, the results will always be the same.

From a point on the surface of the moon the jet flying around the Earth has a relative motion.

If the planet is rotating west to east, does this mean flying either directly east (or directly west) means the flight path would have no arc?

Everything is in motion, you have to pick a reference point or frame, frame being an x,y,z coordinate system in space.

 

[fast]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

 

[bilby]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

The shortest distance between two points on the surface of a sphere is only a straight line if you are allowed to drill a tunnel.

 

[steve_bank]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

I think I see your problem. It is not plane geometry, there are no straight lines in global navigations.

A plane that flies straight is flying a tangent line to the Earth. On a cross country flight the jet maintains constant altitude above the surface. The path is somewhat circular.

Same with a boat. The ocean is curved, In flight navigation it is spherical geometry and spherical trigonometry. On a sphere the shortest distant between two points is a curve not a straight line. It is not plane geometry. In navigation it is called The Great Circle.

Regardless of the spin of the earth, you will always cross the same distance between two points. The difference is the ground speed. If you are travelibng 200mph against rotaion you have one ground speed. Fly with the spin and you have another ground speed.

When taking off a plane will take a heading to a point on the surface. Flight controls are adjusted over time to maintain a heading. When yoyo are flying a small plaint visually you set a heading based on reported winds. Along the way you check for ground references to check your course. You do not consider spin when determining a flight plan. you consider winds.


I had to refresh my memory. East west flights are primarily affected by prevailing winds.

https://en.wikipedia.org/wiki/Great-circle_navigation

Great-circle navigation or orthodromic navigation (related to orthodromic course; from the Greek ορθóς, right angle, and δρóμος, path) is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such routes yield the shortest distance between two points on the globe.[1]

https://en.wikipedia.org/wiki/Airspeed

Airspeed is the speed of an aircraft relative to the air. Among the common conventions for qualifying airspeed are indicated airspeed ("IAS"), calibrated airspeed ("CAS"), equivalent airspeed ("EAS"), true airspeed ("TAS"), and density airspeed.

https://en.wikipedia.org/wiki/Ground_speed

Ground speed is the horizontal speed of an aircraft relative to the ground.[1] An aircraft heading vertically would have a ground speed of zero. Information displayed to passengers through the entertainment system often gives the aircraft ground speed rather than airspeed.

Ground speed can be determined by the vector sum of the aircraft's true airspeed and the current wind speed and direction; a headwind subtracts from the ground speed, while a tailwind adds to it. Winds at other angles to the heading will have components of either headwind or tailwind as well as a crosswind component.

 

[Jokodo]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

The "arcs representing flight paths" are not due to the earth's spin or the planes anticipating where their destination will be, they're by and large due to the fact that the shortest route between any two points is never along a parallel (except when the parallel is the equator), but rather along a  Great_circle, plus some following winds.

Also, the shortest route from a point 500 miles to your east to a point 500 miles to your west does *not* cross your position, and that's true for air routes as much as surface routes. Unless, again, you're at the equator - or if by "a point 500 miles to your east" you mean the point you'd reach in a great circle route starting at your position with an initial bearing of 90.

 

[skepticalbip]

I think people get misled because globes and maps have longitude and latitude lines. Those lines are quite useful for some purposes but meaningless for finding the shortest path across the globe. Also flat maps to depict globes distort reality.

 

[Jokodo]

I think people get misled because globes and maps have longitude and latitude lines. Those lines are quite useful for some purposes but meaningless for finding the shortest path across the globe. Also flat maps to depict globes distort reality.

longtitude lines are *quite* meaninful to find the shortest path between two points along the same meridian.

It is just latitude lines that are useless.

Might have to do with the fact that there is a north and south pole but no east and west pole ;-)

 

[skepticalbip]

I think people get misled because globes and maps have longitude and latitude lines. Those lines are quite useful for some purposes but meaningless for finding the shortest path across the globe. Also flat maps to depict globes distort reality.

longtitude lines are *quite* meaninful to find the shortest path between two points along the same meridian.

It is just latitude lines that are useless.

Might have to do with the fact that there is a north and south pole but no east and west pole ;-)

Except on the equator. In that case, that particular latitude line is on the great circle. However since most trips are not parallel to either latitude or longitude those lines are meaningless for finding the shortest distance for the route. But on the average flat map the longitude lines are shown to be parallel which creates a major distortion that is misleading in determining the shortest route.