Math Question.
 

The following question has been bugging me.


If there is a car at sea level, and an airplane at 30,000ft above sea level keeping a constant altitude and a constant speed of 200mph. Racing from one point to another, who would get there first? I'm thinking the car, because it has less space to travel than the plane.

Does the plane need to go to a point 30,000 feet above the car or does it have to get to the point on the ground the car is at?

If you're implying that the curvature of the earth would make the car's travel shorter, and that the plane follows the same curvature, and there are no other factors, the car would indeed arrive first.
Just like two runners moving at equal speeds around a track but the first is in lane one and the second is in lane 8.
Is that what you meant?


I'm curious as to how this question would be bugging you. :hmmm:

I meant if the point went straight up from where it was on the ground.

@Underseacpl: It was begging me because the two people I asked in my house hold both told me different answers. The question stemmed after playing FSX, I was wondering why airplanes did not from their take off to their landing destination in a straight heading line. And then I was wondering if a plane would get to its destination faster if it were to fly lower to the ground.

It sounds to me like you might be talking about a great circle.
http://en.wikipedia.org/wiki/Great_circle
Since the Earth is curved, the shortest distance from point A to B is not neccesarily a straight heading, but one that circumscribes the curvature as much as possible.
If your car and plane were to go from one point to another, and the plane used a great circle route, the plane would get there faster, because it would be travelling less distance.
If the plane were to try to fly along the same heading as the car the whole time, whilst maintaining altitude, it would arrive later, because it would be going farther.
If they both used a great circle route, the plane would arrive last (still has farther to travel, because it is describing a wider arc)

Whether or not flying lower to the ground would make it go faster depends on a lot of factors, IRL.
If there were no air resistance or anything, and propulsion nuances were not factored in the plane would still arrive first if it used a great circle route. If it followed the car's heading it would still arrive last, but it would not lose by as much as if it had been at 30k feet.

Does that make sense?

Of course, the Great Circle and the situation you describe are not the same. The Great circle in much more complicated, while your scenario can be explained easily using a race track metaphor.

Great circles in a nut shell: Take a piece of string, stretch it between two points on a globe. That's a great circle - the shortest route between the two points.

The thing is, most of the maps we use are Mercator projections - maps where things are stretched out as the lattitude increase, so that angles stay constant - If you draw a line on a Mercator map and it meets meridian 0 at 30 degrees, it will cross meridian 15 at 30 degrees. Parallels and meridians also meet at a convenient right angle. (Mercator maps are also the reason many people are confused as to the relative size of Greenland and Africa.)

Anyway, if you draw a straight line between the same two points you used earlier on a Mercator chart, you get a straight line - What we call a rhumb line. Obviously. Now, if you take the coordinates of the line every five miles, then plot that on the globe, you'll get a line that curves toward the equator.

If you do the same, but copying the line you traced on the globe with the string to the chart, you'll get a line that arches away from the equator.

The great circle is shorter (Since the Earth, SHIV nonwithstanding, isn't actually flat), but it takes you closer to the poles and it's a pain in the rear... And for it to really make a difference, you have to have about 600 nautical miles to go. Otherwise, you're saving pennies.

...yeah, my explanation isn't that great. But in my defense, I had to do 45 hours of spherical trig in nautical school, which is essentially "Great Circle 101" - It's surprisingly easy to explain when you have a blackboard...

all of that aside...

assuming a "standard atmosphere", a jet airliner at 30,000 feet, indicating an airspeed of 200mph would have an approximate "true airspeed" of 323 mph.

assuming the aircraft had selected an altitude of favorable winds, the "ground speed" could be as much as 100 mph faster than either of the above figures, and would be the actual speed of the aircraft over the ground.

however, jet airliners routinely operate at Mach.70 (give or take) even if indicating 200 miles per hour, the ground speed of the jet would be considerably faster... in fact more in tune with a 400-500 mph ground speed.

long story short... in a real life scenario.

the car would have to follow "road miles" at 200 miles per hour, having to slow down for various corners etc

the jet would have to follow air miles at 200 mph.

assuming the trip was to go from LAX to JFK and each respective vehicle had to use navigable routes...

the aircraft would have to travel approximately 2200 air miles

the car would have to travel approximately 2900 road miles

under ideal conditions the jet would make the trip in approximately 6 hours

under ideal conditions, with team drivers, and considering a ten minute fuel stop every 300 miles the car would make the trip in a shade under a day

in a universe where there was no air pressure, no wind, and no planetary rotation.

this would be the case.

see my above post however.

i defy anyone to hop in their car and drive "straight" from LAX to New York :haha:

quite a few signifigant obstacles in between the two.

Just like I asked my first grade teacher......What's the frigging difference.................:har::haha::haha:

Doing ATC entrance exams at the minute, I'm swimming in problems like these.

No more, not on Sunday :)

Back up a minute, is the planes speed measured as air speed or ground speed?

Assuming that regardless of the measurement of speed - if both vehicles have the same RATE of speed (however it is measured) and no obstacles or stops in the way - then yes - the plane takes longer because it has farther to travel.

The reason for this is simple math. If you were to follow the curve of the earth at ground level - from point A to point B - it would measure a certain distance. If you measure that same point A at 30k - to point Bat 30k - maintaining the altitude (accounting for the curvature of the earth) then the distance would be slightly larger.

Draw a circle or "pie crust" and cut it into quarters. If you measure the length of one curved section - it will measure a certain length. Then extend lines PAST the "crust" say 2 inches, mark those points and draw a curve with the same properties as the original - and measure. Its longer. The farther from "ground level" you go - the wider your "circle" becomes - meaning more distance to travel.

Now this again only applies if both vehicles can travel along the curve unimpeded, without stops, and at the same RATE of "real" speed.

Once you start factoring Air vs Ground speed, stops, air density, windage, etc etc then it starts making your head hurt and its just easier to say the plane wins. If it didn't - why would we use them? Of course - planes are simply capable of much greater speeds than cars....

My point was that, if the planes speed is measured in ground speed and the car is allowed to travel in a straight line with the same frictional constraints as the plane, then they would be covering ground at exactly the same speed (definition of ground speed) and would, therefore, arrive at the same time.

The real problem is that the question is unanswerable in its current form because it doesn't give us enough of the factors needed to reach a conclusion.

Not if one builds a tunnel.