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Suppose one needs to travel to Somalia from Indonesia(which are approximately at a distance of $4000$ miles from each other on the line of equator). To accomplish this task, I suggest the following method:

Design an airplane that can surpass the earth's atmosphere radially(~ $300$ miles thick) and stand still in the air till the landing point of Somalia is just about to arrive radially beneath the plane's stationary point(earth rotates from west to east) at which time instant, the landing process is initiated. In this way, the total time taken would be:

\begin{equation} T=\frac{4000(mi)}{1000(mi/h)} + \epsilon \end{equation} Here, $1000(mi/h)$ is speed of any point on earth's equator and $\epsilon$ is the small time the aircraft took to travel to and fro through the earth's atmosphere.

The total approximated journey time with this method would not exceed $6(hr)$ which is considerably smaller than the actual time taken of about $12(hr)$. My apprehension is why this technique is not being used to facilitate travelling between two different points when the absolute speed of the aircraft(w.r.t space) can be reduced to zero by surpassing the earth's atmosphere?

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  • $\begingroup$ Note that the atmosphere is only about 60 miles high, for practical purposes. (Above that, drag affects satellites at orbital speed on a scale of hours, days, or weeks, rather than the minutes or seconds at lower altitudes.) But I'm not clear how you're "standing still", exactly. $\endgroup$ – Nathan Tuggy Feb 3 '16 at 3:26
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Summary: It's not a great idea. It will take you over 8 hours, and you'll end up 5400+ miles above your destination. Details follow image.

enter image description here Since you don't specify the speed of the plane, let's assume for the moment that you can move 300 miles up from your current location instantaneously.

To avoid the air/wind problem, let's also assume we are doing this on an airless planet that is otherwise similar to Earth.

Finally, since you can "stand still in the air", we'll ignore the effect of gravity as well.

Since your starting point is rotating, its (x,y) position in the diagram above is modeled as:

$ \left\{\text{eer} \sin \left(\frac{2 \pi t}{\text{sidday}}\right),\text{eer} \cos \left(\frac{2 \pi t}{\text{sidday}}\right)\right\} $

where t is the time in seconds since the "launch", eer = 6378.137 is the Earth's equitorial radius in kilometers and sidday = 86164.1 is the length of the sidereal day in seconds.

And your destination is:

$ \left\{-\text{eer} \sin \left(\frac{\text{dist}}{\text{eer}}-\frac{2 \pi t}{\text{sidday}}\right),\text{eer} \cos \left(\frac{\text{dist}}{\text{eer}}-\frac{2 \pi t}{\text{sidday}}\right)\right\} $

where dist = 4000*1.609344 is the distance to your destination in kilometers.

Since your launchpoint has an initial velocity of about 1000mph, so do you. This means your position at time t is:

$\left\{\frac{2 \pi \text{eer} t}{\text{sidday}},\text{eer}+\text{hi}\right\}$

where hi= 300*1.609344 is your initial height in kilometers.

With these conditions, you will be over your destination about 28911 seconds after you start (8 hours, 1 minute and 51 seconds), at a height of about 8718 kilometers (about 5417 miles).

Your average surface velocity would be right around 500 miles per hour, slower than a supersonic airplane, and even slower if you include the time to ascend 300 miles at the start and descend 5417 miles at the end.

Of course, if your magic plane can move 300 miles in time $\epsilon$, you could just as easily apply a westward thrust of ~1000mph to cancel our your initial eastward velocity, in which case your calculations would be correct.

A plane as powerful as this, however, could probably travel anywhere in the world rapidly, without help from the Earth's rotation, so you'd be best off aiming it west, just as you would a normal airplane.

I toyed with the idea that you could launch at a high velocity and land back on Earth at your destination purely due to gravity (albeit at a very high speed) as per:

Trajectory of projectile launched from planet's surface

but haven't been able to get the numbers to work. My work on the high-velocity launch idea is at:

https://github.com/barrycarter/bcapps/blob/master/MATHEMATICA/bc-solve-physics-232844.m

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The first problem is just that there is no free lunch in decelerating/accelerating; in particular, there's nothing in space to slow you down, nothing that is "unmoving" in any real sense to push against, so if you go straight up, you're still moving along with the earth's rotation at the same speed. (Not the same angular velocity, but the same mph.) So just going up and then coming down doesn't buy you all that much: you cannot "reduce absolute speed to zero" that way without actually reducing speed in exactly the same way you normally do (by burning fuel).

By going outward, you can reduce your angular velocity somewhat, meaning that you'd no longer quite keep up with the number of (micro)radians per second that the earth is traversing. Unfortunately, the earth's radius is a good few thousand miles, and going a measly few hundred out will do very little that way.

A second problem is the challenge of hovering for that long; you're not going nearly fast enough to use orbital mechanics to fall around the earth continuously for free, so instead you have to burn fuel, and a lot of it, to keep up a continuous 1g vertical acceleration just to keep from falling. You can save a bit by simply accelerating in a higher arc and then letting yourself drop down. That way, gravity is reduced a little at the top. But it takes quite a distance before that saves you much, and if you go too far, you'll take too long to fall back down and you'll land in the wrong place.

The third problem, although far smaller, is that to the comparatively simple challenge of keeping airliners pressurized you have now added the thornier problem of maintaining atmosphere in space for hours. Worse, you also have to manage temperature. That's harder than it sounds in space, where there's glaring sunlight and no air to carry away heat. (With hundreds of warm bodies in a small enclosure, freezing would not be a problem.)

Finally, you have to use rockets for a lot of this. Rockets are less efficient than jet engines by a large margin, because they have to carry their air with them, and that requires extra fuel and extra oxygen, which requires more fuel and oxygen, and so forth. When you're also doing far more acceleration than a normal airliner, the result is burning many times as much fuel, which also requires a much larger (and more expensive) vehicle to carry all that.

Oh yeah, and you're probably going to be breaking the speed of sound at various stages during your flight. This is already a practical problem for airliners; Concorde was retired largely because of problems getting clearances where people wouldn't be bothered too much by sonic booms, and the fuel flow at supersonic speeds is ... you guessed it, much higher than subsonic, where airliners normally are.

In sum, fuel costs are going to be perhaps hundreds of times as much, the vehicle will cost far more to design and build, you'll have to train pilots in new and exciting ways to handle problems... and you won't actually save all that much time, if only because you'll have to fly complicated paths around anywhere that's populated.

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