I don’t do conventions very often, but I recently went to ConBust out in Northampton, MA, while visiting some friends. While I was there, I had a guy propose something fascinating to me. I can’t remember the guy’s name, so if he or one of his friends sees this, post your info in the comments. (Edit: it was a dude by name of Thom Howe.)
The guy Thom had an idea for a date. He wanted to rent a cherry picker, drive it to her door, and pick her up in it.
Then, he’d drive to the beach, and get there at just the right time to watch the sun set.
Once the sun had set, he’d activate the cherry picker, they’d be lifted up above the beach …
… and they’d watch the sun set again.
Clearly, this is an excellent idea, and any girl would be lucky to see this guy Thom at her door. But is it plausible? How fast and how high does the cherry picker have to go?
I tried to work out the answer for him there at the table, but there was a line of people and there wasn’t time. But when I got home, I remembered it again, and I’ve worked out the solution.
Here’s the situation:
By the time the earth has rotated through angle theta, the cherry picker will have to have climbed to height h.
After t seconds, theta in radians is:
The height of the lift above the center of the earth is:
So the height above the surface (sea level) is:
Substituting everything so far we get this expression for the height the lift needs to reach t seconds after sunset to stay even with the sun.
Now, an actual cherry picker has a maximum lift rate (I Googled some random cherry picker specs, and 0.3 m/s is a normal enough top lift rate.) We’ll call that rate v, so the actual height of the lift will be this:
Substituting that in and solving for v, we get this:
(That’s arcsecant, not arcsecond). This equation tells us how fast the lift has to go to get from the ground to height h in time for the sunset1.
But we can also get the answer by just trying a few different heights. We plug it in to Google Calculator2:
2*pi*6 meters/(day*arcsec(6 meters/(radius of earth)+1))
and find that h=6 meters gives about the right speed. So, given a standard cherry picker, he’ll get his second sunset when they’re about six meters up, 20 seconds later.
You might notice that I’m ignoring the fact that he’s not starting at sea level — he’s a couple meters above it. This is actually pretty significant, since the sunset line accelerates upward, and it brings down his second-sunset height quite a bit. If he got a faster lift, or used an elevator, the correction would become less necessary. Extra credit3 for anyone who wants to derive the expression for the height of the second sunset given the lift speed and height of first sunset. For now, I recommend he dig a hole in the sand and park the lift in it, so their eyes are about at sea level4.
1 Ideally, we’d solve for h, but it’s inside the arcsec and that looks like it’s probably hard. Do one of you wizards with Maple or Mathematica wanna find the result?
2 If you work in one of the physical sciences and don’t use Google Calculator for all your evaluatin’, you’re missing out. I wish there were a command-line version so I could more easily look/scroll through my history. I know Google Calculator is largely a frontend to the unix tool units, but it’s better than units and available everywhere.
3 Redeemable for regular credit, which is not redeemable for anything.
4 I suggest a day when there aren’t many waves.
oh… and funny comic.
that would be terrible, though, if the guy showed up in a cherry picker and told her his idea, and the girl just said, “hey, isn’t that from xkcd?” this blag reaches a huge audience but hopefully it’s all guys.
On the contrary, if that happened to me (as a female) I would say “Hey, isn’t that from XKCD?!” and be really excited about it.
I haven’t yet worked out the math, but I beleive there is also a correction for latitude that needs to be done. Unless you are at the beach on the equator – which would be a very good place for a date. I think the latitude correction would only effect the time available rather than the height required.
I don’t think refraction will play a very big role in the decision as h is small enough that refraction would be so minusculely (that’s right, made up adjective) different that it would not be noticeable to the naked eye. Light is still refracted by the Earth’s atmosphere even at 6 meters above sea level. Heck it’s probably not even noticeable until you are above 100 meters (I don’t know what the actual height might be because that is based entirely upon perception).
This is similar to the fact that the Earth is actually an ellipsoid (its bend being so little that it is unnoticeable).
Wait. I’m probably wrong…maybe refraction would be noticeable. Anyone with a huge garage happen to own a cherry picker? And live by the ocean? And follow this blag? It would save us some theorizing.
In the Badlands of South Dakota, my friend Mary and I saw three sunsets in one day. We watched the sun go over one rise, ran up to the top of it, watched it go over a second rise, ran to the top of it, and then watched it go down for a third time (when I took possibly the best picture that anyone has ever taken, of Mary, hands to the sky, between me and the setting sun).
A few notes:
a) dear thom: watch it with the intellectual theft, young man.
b) …. but thank you for bringing this to randall’s attention, something I wouldn’t have gotten around to even if i had been back in northampton.
c) so, um, that’s actually neither a cherry-picker nor a forklift. it’s a vertical lift. a forklift has two prongs and lifts pallets, and a cherry-picker is an articulated-boom lift that sits on the back of a truck.
d) as i said to thom many years ago… i will never forgive myself for the one time i drove by a used-car place that was displaying a cherry-picker for sale, and didn’t buy it – the fact that i was 12 then notwithstanding.
e) other advantages: when picking up said lady, extend the boom horizontally right up to her front door, and knock. no walking = fucking sweet. also, the bucket designed for a single person means she has no choice but to snuggle up. finally, driving a beast of a greasy, ugly, diesel truck makes you feel manly.
I was flying to Chicago a few weeks back and was flying due northwest, I think. Out of my window I could see the sun setting over the cloud horizon, towards the front of the plane. It was very pretty. I might’ve been imagining it, but I kept thinking that the sunset was abnormally long (an hour or so) because we were flying west, chasing it around the earth. Is that possible?
Good idea but only really plausible on the west coast. Otherwise if you show up for a date at 5 AM with a cherry picker, a young lady would not be as open to the idea to watching 2 sunrises.
Hmm.
I got h = 5.3364030833514578685474222761059
Sound about right?
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I go gliding, and one of our group once managed to see the sun set twice in a day by watching it on the ground, taking a winch launch, and watching it again.
A winch launch can get you to 1,000ft in a matter of 10 seconds on some days, probably the fastest thing in the world that can get you to 1,000ft from a standing start. Even the space shuttle can only just manage that. In any case, it’s more than is required according to your calculations, although the airfield was at 220ft or so above sea level.
Other things you can do while airborne include seeing the entire ring of a rainbow, no pesky horizon to get in the way!
You have to be careful if it’s a hydraulic lift device.
Otherwise you’ll get hydraulic leakage (after picking the couple up, it will slowly leak down). It depends on the hydraulics and the weight involved.
Such a scenario, involving a hydraulic lift used for roof access, got a friend of mine stuck on a roof for hours.
As I saw the first baloon I thought it was going to be a play on word on popping one’s cherry… Glad there is some romantic people left out there in the world!
@Blake: Your friend was probably using the wrong piece of equipment, then. The only hydraulic equipment designed for providing access is an elevator; most of those aren’t even actually hydraulic anymore.
Save yourself a lot of hassle; Get a ladder, a forklift, and another person for your roof work. Also, don’t do anything potentially dangerous (like working on a roof) alone; A lot of injuries that start minor (e.g. Heat exhaustion) can get worse if left untreated, especially when combined with heights.
man, you are crazy and i love it xD
Sorry but your math is horribly flawed. Let us continue from:
h = r*(sec(2pi*t/day)-1)
to simplify it, w=2pi/day
h = r*(sec(wt)-1)
for small values of wt, sec(wt) = 1/(1-wt)
h = r*(1/(1-wt)-1) = rwt/(1-wt) || *1/dt
v = rw/(1-wt)?2
For the time frame we are intrested in, the speed of the point where sun is observable is moving upwards at speed of 460m/s. Sorry guys, it would have been awfully romantic but with cherry picker (w/o rockets) it is not going to work.
What, no adjustment for refraction when light enters in the atmosphere?
[...] Need Not Be So. Alerted via a mention on XKCD, bolstered by the acumen of dozens of dimensionally aware readers***, I discovered that a [...]
The sand, and hence the air just above it, is going to be hotter than the air 20ft up. The air will therefore be less dense at sand level, causing the light to refract upwards, causing the sun to disappear before it reaches the sand-level tangent. Going up 20ft may make the sun visible for this reason as well as the change in tangent.
I worked out the following:
When Y is the height in meters when you see the sun exactly at the horizon (let’s say the sun is a point),
X is the time in seconds since Y was zero, you get the following:
Y = (6371 / cos ( (1/240) * X ) – 6371 ) * 1000.
When you are at 18 seconds, you need to be at six meters to see the sun exactly at the horizon. I think this resembles his calculations very closely :) and also gives you a nice graph to plot ;)
In my example I used 6371 as the earth’s radius, and calculated that each second the Earth turns 360/24/60/60 =1/240 degrees. The ” * 1000″ at the end is to convert the kilometers to meters. Forgot to tell all this.
If you take a flight in the right direction at exactly the right time you can see the sun rise in the west. A much more expensive date.
you make math so much more interesting for some reason. :( I fell asleep through algebra II at least 3 times a week.
Some girls would slam the door in your face if you arrived in a cherry picker. I mean – if you bring heavy machinery to “pop her cherry”, you’re doing it wrong…
Sounds like a lot of work just to get ~20secs extra sunset, at least half of which will be in either “pause” or “slo-mo reverse”. Still, if she doesn’t see it as insanely geeky (in a bad way), it might come across as endearingly romantic …. ally geeky.
I never knew those things were able to drive themselves, at least not at more than sub-walking pace. I thought they had to be towed by truck/pickup/SUV, or on a flatbed.
Owen: Every day on my drive to and from work, I pass a large industrial hire yard that has about 30-40 of these things in different sizes and colours, including several big enough to rise above the elevated bit of motorway I’m travelling on (which is higher than pretty much everything else except clocktowers). On days with low wind speeds they crank them all up to random heights with advertising placards attached. If you’re still serious and fancy a journey I can give you the address ;)
derive the expression huh? To nonmathmatically derive for humour:- well it goes from “you’re so sweet” to “I’m going to love you forever”.
Alternately, the height can be expressed in how many heartbeats at what rate of acceleration due to stimulation of the “awww” factor. If she’s not a geeky kind of girl that is probably a ho-hum 30 beats.
But if she IS a geeky and romantic lover of “Princess Bride” kind of sweetheart then you just may find yourself making the second sunset at the 50 beat mark. Lucky you !
Anyone knows how i can determine speed from the function i wrote? It must be possible, I just need to know what the f ‘ (x) is…
@Mikko:
There has to be somethign wrong with your math, because if the sun’s observable point is moving by 460 m/s vertically, matt’s story would be impossible. Perhaps it moves 460 m/s laterally, which still gives us an acceptable window.
Also, the formula solved for h is simply:
h = r[sec(2*pi*h/[v*day]) -1]
Further, the life would actually have to go a bit higher in order to watch the full sunset again. But it all seems reasonable and quite a good idea to me.
did anyone consider snell’s law?
Doesn’t all this math assume that the sun is a single point? Moving up 6 meters just to watch the last tiny bit of the sun dip below the horizon is not exactly watching it set again.
I believe the sun has an apparent size of half a degree, so get even half of it above the horizon it seems to me you would have to go up alot more than 6m…
After reading the ideas, and the comments, I would suggest an alternate approach for about the same price and promising to be a lot more romantic.
Rent a room – a meeting room facing west so she doesn’t get the wrong idea. On the 20th floor of a high rise with a roof top access.
Watch the sunset out the window (don’t care about the waves or the closeness to the equator!
Walk, don’t run, to the elevator to the roof and leisurely watch the sunset again!
Just use a helicopter. You can sit in it, watch the sunset and then rise several times to watch it again. The price of renting one would be high, but cherrypickers aren’t cheap to begin with.
or you can rent a balloon accelerating in synch with the time and watch the sunset in stasis in the horizon
omg :O
This must be one of my favorite posts. I’d love to go on a date like this.
@Me, Myself, nor I:
f’(x) = ( 159275 / ( 6*cos( x/240 )^2 ) ) * sin( x/240 )
I’ve tried this in a plane before, so I know it works, but I would love it if someone would do the math to figure out how moving laterally at about 100 knots affects the time and height required. The time it didn’t work I think was because we were heading east. I hit a climb rate of greater than 1300 feet per minute for about 15 seconds, then about 500 fpm after that and wasn’t able to get the sun to come up again.
Well how about simply including an elevator in the equation. Rater than a cherry picker, have a date next to a skyscraper. Then after the sunset, go up to the observation deck and do it all again. This would probably work best in a city like Dubai.
So, I live in an hour from Boise, Idaho right now. Anyone have any ideas for cities with the nearest skyscraper near the coast? I have resolved to try that idea.
A big flaw in this plan is the fact that for OH & S reasons, if you’re going over 2m high you need to be wearing a harness. Now those things are just not sexy, even if you’re into bondage.
funfact!
well, due to the refraction of the earth’s atmosphere the sun actually sets below the tangent. (well, from the sun to you) and you see it set roughly 5 minutes after is already has.
doesn’t really affect the problem too much though.
Well, in this case the experiment is much easier than the math. One simply needs to observe the sunlight on a tall structure, like a building, a week or so before. There are many beach houses that would serve this purpose (and taking her to a beach house would make a much nicer date, anyway.)
OK so the problem for me is that when we think of a sunset we don’t think of an instant in time. THis is especially true at the ocean. What happens is that the sun appears to sink into the sea like its a submarine. This takes time to occur. The Cherry picker could prolong the sunset but it can’t create two distinct sunsets. To do that you need to do something like what matt said and watch the sun set over a rise and then climb to the top of it. Also if you are looking for a really prolonged sunrise I suspect that you could go to the Artic circle at the right time of year. I know that there are a few days in the year where the sun doesn’t set at all. But I suspect there is a location and path that you could take that would result in a very prolonged sunset.
Aren’t you forgetting latitude? I’ve seen the sun set in the tropics, and I’ve seen it set near the arctic circle. The first takes minutes, and the second takes hours.
I have been told by someone who saw it more than once that, if you took the Concorde from London to New York at the right time of year, you could see the sun set while you were on the ground, then rise as the plane rose to cruising altitude, and then set again. Not quite the same as the cherry picker on the beach, though.
There’s a similar idea in Julian Barnes’ novel “Staring at the Sun,” involving airplanes and rapidly climbing or descending to see a sunset or sunrise twice.
Seems like you could do this in a tall enough building given a skyline that is conducive to seeing a sunset. Perhaps a Miami beachfront condos with one of those external glass enclosed elevators?? (other examples exist, I’m sure)
Okay… Suggesting to spend incredible amounts of dollars to rent a cherry picker and pick her up. Definitely not early dating material, definitely for someone very deserving who have demonstrated she is worth something worth that much effort.
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