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Separation

Jacob from NY writes:

“Alex, what’s the answer?”

Dear Jacob,

Thank you for your question! Separating can be difficult for many people, so I am happy to have the opportunity to write on the topic.

First, let us assume a spherical Earth.

For the sake of simplicity, let's assume that the boy is crying due North along the Prime Meridian and that the girl is walking East along the Equator.

Let us next define our axes. We shall let the x axis be the initial (easterly) direction of the walking girl and the y axis be the (northerly) direction of the heartbroken boy. We will let the z axis be “up” (from the ocean) at their starting point. For simplicity, let’s make the center of the Earth the origin point. So defined, their initial position is (0, 0, 1) also known as “on top of the world”:

This means that they are both fleeing in unit circles, just in different planes. The girl is traveling in the circle that seperates the Northern and Southern Hemispheres (the xz plane):

And our heartbroken boy is traveling five times faster along a circle in the yz plane, which cuts the Eastern and Western Hemispheres:

Recall that the parametric equations for movement along a circle are:

So, the specific equations for their locations at time t are given by the functions:

We can get the distance between them in each dimension simply by subtracting:

The total distance between two objects in three dimensions can be found using the same Pythagorean Formula (also called “The Euclidean Distance Metric”) used in two dimensions, just with an added term for dz. This states that the distance s separating them satisfies the equation:

So, plugging in the equations above, gives us the distance between them to be:

At this point it’s worth taking a step back and sanity checking what we’ve found. We have found that the distance between them is periodic. The cosines above have a range from [-1, 1] so we can get a distance anywhere from 0 to 2 R. This makes sense, as any two points on Earth are, indeed, between “zero” feet and “The Diameter of the Earth” feet away from each other.

This formula also shows that there is no distance between them at time t=0, as one would expect.

So let’s plug in the actual values and see how far away this equation claims they are from eachother at t = 5.

First, let’s get an idea of what we expect the value to be. A bird’s-eye view at t = 5 shows through simple trigonometry that we should expect the distance to be appropriately 25.5 feet:

So let’s find our real value.

The circumference of the Earth is 131.48 million feet. Traveling at 1 ft/sec means that it will take our heartbreaker 131.48 million seconds (4.2 years) to circumnavigate the globe. Traveling five times faster, the boy takes only ten months per trip. Plugging these values in to our angular velocity equation, yields:

Plugging these functions in for the angle in our distance function yields our final distance function.

Now that we have our distance function, let’s test it out! Plugging in t=5 gives us a distance s of …

very close to the 2D approximation of 25.5! With only 0.1% difference between the two, this is a reasuring sign that our function is accurate.

For the first hours, we expect the distance between them to grow linearly, as the curvature of the Earth won’t yet have much effect on their separation. Plotting our distance function for the first hour of separation shows exactly this linear trend:

But what about after this initial period of time? Recall that after ten months of travel, our heartbroken boy has circled the whole globe. What does the distance between them look like over that stretch of time?

After nearly five months of travel, the boy has reached the other side of the planet where the distance between them reaches its peak. His journey continues and eventually brings him back to where they broke up. But, of course, by then she has moved on and is nearly 5000 miles away:

one fifth of her own way around the equator.

So our distance function works and tells an interesting story. But we don’t just want to know the distance between them. The question asked how quickly they are moving away from each other.

To answer this, we need Calculus.

In order to get a rate, one must differentiate a distance function with respect to time (to get “change in distance” over “change in time”). For our separation problem, we must differentiate our Euclidean metric. Doing so yields:

Note that tiny s in the denominator. The derivative of the distance function (ds) is a function of the distance function (s)! This is a bit strange at first but perhaps it should not be so surprising that distance has an effect on how we move in relation to each other.

Like we did earlier, it’s a good idea to sanity check our work against an approximation.

In the 2D model, the rate of change of the distance is found using the same basic approach, but the math is a bit simpler:

Recall our 2D diagram from before (this time with the velocities added):

Plugging in our values into the above partial equation yields:

So 5.099 ft/s is the answer according to the 2D approximation. But what does our 3D model say (keeping in mind that over this short time span the curvature of the Earth should not matter much)?

So, to answer your question, five seconds after seperating they are moving away from each other at a rate of 5.093 feet per second.

But how fast are they moving away from each other further on? Recall that they actually started to move closer to each other after about five months. What does that look like?

So we can see that after the initial period of running away from each other at just over 5 ft/s, eventually their movement starts to become more orthogonal: drifting closer and further at a more liesurely rate of about 2 ft/s.

But what if we zoom out even further, and look at their next few years?

What’s that?! It looks like after a couple years there’s a discontinuity where they start coming back together only to be (instantly!) running away again. What does the distance function look like over this period?

It’s a collision!

Recall that the boy is traveling five times faster. After 25 months, our crying boy has circled the globe exactly 2.5 times and so is halfway through his third circuit, having just arrived back at the equator by way of the North Pole. To his amazement, the girl has, over that same time, meandered exactly 0.5 times around the equator.

And so they meet again. Years and literally a world away from where they broke up, their distance s comes back to 0.

And while they are only together for an instant (assuming they continue on their seperate ways), for that single moment (because the null distance s adds a zero to the denominator of the derivative) they cannot be said to be moving towards each other. Nor can they be described as moving away from each other. Nor are they standing still.

That moment when they meet again is simply, mathematically, indescribable.

Best wishes from Thailand (s=3.7342×10⁷ ft),

Alex

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