# Thread: Which way do I go

1. ## Which way do I go

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?

2. *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.

3. 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.

4. 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.

5. Originally Posted by 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.

6. Originally Posted by fast
Originally Posted by 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?

7. Originally Posted by 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.

8. Originally Posted by fast
Originally Posted by 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.

9. Originally Posted by steve_bank
Originally Posted by fast
Originally Posted by 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.
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?

10. 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?

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