Two Planes Keep Flying

Two days ago I threw at my readers the following problem:

A plane takes off and goes east at a rate of 350 mph. At the same time, a second plane takes off from the same place and goes west at a rate of 400 mph. When will they be 2000 miles apart?

The purpose of throwing this problem was to discuss the nature of the implicit assumptions that we are asked to make when solving math problems, and the implicit assumptions we teach our children to make when we teach them to solve math problems. This is especially important for problems like this, that are phrased in terms of a situation in the real world. The real world is too complex to model all of; the great power of mathematics is that sufficiently idealized situations are predictable. But which idealizations are appropriate? How does one choose? How does one teach youngsters what to choose?

Before I get to the actual discussion, however, I want to re-throw this problem at my readers, in an effort to highlight what originally jumped out at me as being wrong with it.

Neglecting the effects of altitude, differential wind, acceleration, relativity, measurement error, finite size and non-superimposability of the planes, and the Earth’s deviations from perfect sphericity,

  1. Find how much time it takes them to become 2000 miles apart, assuming that the planes are starting from Boston and the distance is measured as
    1. a straight line in 3-space.
    2. the shortest surface distance.
  2. How far from the closest pole may the starting point be located, so that the answer to the problem is “never”? Solve separately for
    1. the 3D distance.
    2. the shortest surface distance.
  3. What portion of the Earth’s surface do the “never”-locations of the previous question occupy?
    1. under the 3D distance?
    2. under the shortest surface distance?

Hint: The easiest question is 2b.



  1. Anonymous Rex:

    I get that the answer to 1b differs by less than 1% from the naive estimate. Is that correct?

  2. Christ Schlacta:

    I’ll admit I’ve only completed through trig and only in a community ollege, so my math skills are a little lax, but are we solving, for 2 a,b, the linear and 3d distances of the planes is never 2000, or the linear and 3D distances from the poles with some assumption about the plane’s distances ?

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