Help with the mathematics of my thought experiment

The other day as I was walking west towards the setting sun, I got to thinking, my walking against the rotation of the earth would extend the length of daytime that I am perceiving. It got me wondering, if I were to start walking west at sunrise, and continue walking until I observed the sun setting, how much longer would my perceived day be?
This led to me wondering about travelling at varying speeds, both east and west, and how that would affect the length of my perceived day.

Before we continue on this journey, I have made a few assumptions:

  1. The observer is at the equator.
  2. The circumference of the earth is 40,000 km.
  3. The rotation of the earth is 1667 km/h.
  4. 20,000 km of the circumference of the earth is experiencing daylight at any time.

Now, imagining a graph of perceived daylight, with the x-axis representing my velocity to the west, and the y-value representing time, I came to the following conclusions:

  1. At x=0, y = 12 hours.
  2. As x approaches the rotational speed of the earth, y approaches infinity, as, if your velocity completely counters the rotation of the earth, you would be forever in daylight.
  3. As x goes beyond the rotational speed of the earth, y comes back down, and approaches zero for extremely large values of x.
  4. As x moves towards large negative values (high speeds east), y also approaches zero.

This got me wondering how to graph this function. It would look like a variation of y = 1/x^2 , adjusted to meet the above conclusions. I have given numerous hours of thought and pen-to-paper effort trying to find the equation that would graph this, but my calculus prowess is not what it once was (or ever was).

Playing around with the graphing function on google, the following equation seems to meet the conditions above, as well as pass the basic thought test for airline travel speeds:

y = 20000km * 1667km/h / (x-1667km/h)^2
y = 33340000/(x-1667)^2

So my questions are thus:

  1. Does this equation represent my thought experiment?
  2. Can someone explain how we come to that equation?

The length of each period of sunlight for prograde speed v is just:


Bingo. Thank you!

I was trying to calculate the time based on the amount of ground that passed underfoot, which increased as your speed increased, and thus time at speed, when I should have been looking at net distance from the dusk point to the dawn point, regardless of the earth rotating through those points. Analogous calculating how long I will stay on a 10 ft treadmill going 25 ft/s when I’m only walking 5 ft/s - it doesn’t matter how much belt passes underfoot, only how long until I’m thrown off the end…

In fairness, plotting the right and wrong answers together looks pretty close

I find that the simplest way to find formulas for things like this is to look at the extreme edge cases: For instance, if you walk at the same speed as the rotation of the Earth, the Sun will never move in the sky, so your day length will be infinite. Thus, the correct equation must be one that goes infinite at that value.