James Pine Has A Blog

During a recent United flight from San Francisco to Maui, I played the “Halfway to Hawaii Game”. The objective, determine the exact time the airplane would reach the geographical midpoint of the flight to Hawaii. Shortly after takeoff, the crew provided this flight data:

Departure – 6:07a Hawaii time
Duration – 4hrs 42min
Distance – 2076mi
Cruising speed – 485mph
Headwinds – 30mph during the first half and 25mph during the second

Participating passengers were asked to make their estimate accurate to the second (for tiebreak purposes), nevermind the fact that neither the departure time, nor duration provided that level of granularity or that pilots are human and don’t necessarily stay true to the flight plan or weather predictions are not very accurate etc The first thought that crossed my mind was, why such a low top speed? The speed of sound through air at sea level is 768mph and I am pretty sure the last few flights I’ve been on all had top speeds of 500+mph. Airlines have been slowing down flights to save fuel for a while now, but 485mph seems a bit extreme.

Getting back to formulating a guess, in addition to all the issues listed above, clearly not enough information had been provided e.g. what was the rate of acceleration/deceleration during takeoff and landing? Other information had been pre-processed in some manner e.g. the headwinds were likely averages over each half of the flight. Using just the duration, one could of course come up with a naive guess figuring the first half of the flight would take 2hrs 21min. However, one might suspect the headwind data should be taken into account as they were supposed to be stronger during the first half of the flight. If we’d be travelling at 455mph during the first half vs. 460mph during the second, ignoring take-off and landing, it would take 2.281hrs to cover the first half and 2.257hrs for the second. The total flight time was supposed to be 4.7hrs so the remaining 0.162hrs needed to be distributed.

At this point, I could have just split the time 50/50 yielding a first half travel time of 2.362hrs or 2hrs 21min 43.2sec, but my gut told me that going from 0-cruising speed happens faster than cruising-0, maybe something like a 40/60 ratio. So I only added 40% of the 0.162hrs to the first half time and ended up with 2.346hrs or 2hrs 20min 46sec yielding a guess of 8:27:46a. The actual time? 8:27:47a (typical programmer off by one error?). No one else’s guess was closer than mine, so I won a Peace Love Ukulele CD by Jake Shimabukuro. I wonder if anyone just halved the duration as that would have been pretty darn close, 8:21:00a, likely a winning guess if it weren’t for someone like me completely over-thinking the problem and probably getting lucky in the process.

When I got home I checked the Internet to see what others had said about this game and didn’t realize United had been a “thing” on United for many years (message board posts asking how to win date back to at least 2002). Apparently past prizes have been bottles of champagne or 3 cans of macadamia nuts; sometimes tailwind in addition to headwind information is provided and/or the distance is given in knots vs mph. People have posted their formulas online, most of them don’t try to account for takeoff or landing but do take into account windspeed. On this flight, those would have resulted in a worse guess than just halving the travel time. Anyway with one win in one attempt and only being off by one second, clearly my formula is the best right 😉

s1 = speed – first half headwind OR speed + first half tailwind
s2 = speed – second half headwind OR speed + second half tailwind
t1 = distance / 2 / s1
t2 = distance / 2 / s2
t3 = total time – t1 – t2
first half travel time = t1 + .4 * t3
first half arrival time = first half travel time + departure time

Note the first half travel time will need to be converted to hours/minutes/seconds. And because no one really cares about this stuff and really just wants to see a photo from Hawaii, here’s a Maui sunset taken via a Nexus S.