I’ve been coming to Hawaii with my family (at Christmastime) for 10+ years, and every year United plays the “Half-way to Hawaii” game. The point of the game is to calculate (or guess) the time that we will reach the half-way point between San Francisco and Lihue, HI (on Kauai). I’ve played a number of times before and remember always giving up in frustration, which resorted in submitting bad guesses. This year I actually calculated an answer, AND I WON! (My guess was within 5 seconds from our actual half-way point).
I can’t say my calculations were advanced or profound (or even correct), but I’ll share them here. I am fully willing to attribute my win to luck (since there actually are a lot of variables involved), although the estimate I submitted this year came purely from my calculations.
Here’s the information the pilots give you:
The total distance to Kauai: 2173 nautical miles
The average airspeed: 470 knots (1 knot = 1 nautical mile/hour)
The average wind speed (head wind): 13 knots
The departure time: 8:37 am (Hawaii Time)
The flight is always about 5 hours, so halfway is around 2.5 hours from takeoff. (I use this as a ballpark to make sure my guess is within reason). Our halfway point this time is 1086.5 miles.
When I thought about our flight path, I realized that it takes some time to ramp up to 470 knots, and in the first few minutes, we are climbing to higher altitudes (so we aren’t covering as much ground). The first thing I decided to do was estimate our distance traveled in the first 10 minutes of flight.
I assumed that in the first 5 minutes, maybe we average 150 knots. In the second 5 minutes, maybe we average 250 knots. If that’s the case, then we travel 12.5 miles & 25 miles in each of the first 5 minutes windows. (E.g., 5/60 * 150 knots = 12.5 miles).
Our average airspeed in the rest of the flight will be 470 – 13 (to account for wind resistance) = 457 knots. Therefore, in 2 more hours, we’d cover 914 miles. At this rate, we’d still have 135 miles left to go (1086.5 – 914 – 37.5 = 135 miles).
Those remaining 135 miles would be flown in (135/457 * 60) = 17.724 minutes. Finally, 0.724 * 60 = 43.44 seconds.
Therefore, we’d reach the halfway point after flying for (pulling my many calculations together): 2 hours + 10 minutes + 17 minutes + 43 seconds. Converting this to Hawaii time brings me to 11:04:43.
Of course, there was some estimating in the first part of my calculation (and our departure time was only listed as hours:minutes). Windspeed and airspeed are only estimates–averages–and are likely to change during the flight. Still, somehow my calculations + luck made my guess only 5 seconds longer than our actual half-way time! Actual time: 11:04:37!! And for this, they gave me a bottle of wine from first class 
