View Full Version : Maths problem!

7th May 2013, 20:21
OMG! my 15 year old has just asked me this question:
After surviving a crash in the desert somewhere near the equator. You get up at sunrise and start walking towards the sun. You continue all day walking towards the sun until sunset, then you stop. How far from the crash site have you gone? At what time of the day were you farthest from the crashed aircraft?
I think I know the answer but only by guessing. :bored:

7th May 2013, 20:29
You ain't goin' nowhere, vee-tail.

7th May 2013, 20:30
You start the day walking East, you slow down as the sun rises until you stop when it is directly overhead (and at that point you are furthest from your aircraft). As the sun passes overhead, you turn 180º and follow it towards the west as it sets. You are probably walking slower at the end of the day than at the beginning - so you are probably a bit short of getting back to the aircraft ;)

The question of "How far have you gone?" depends on whether you are talking distance or displacement ;)

7th May 2013, 20:32
I would suggest that if after surviving the car crash you walk towards the sun instead of seeking help/shelter you deserve all you get.

But in Maths terms WGF=WGF/D

Big Hammer
7th May 2013, 20:33
Having a few drinks under the belt and not giving this much thought I would say you are furthest away at noon and by sunset you would be back at the crash site. i.e. sunrise in the east, you follow the sun until it is overhead at midday and follow it back until it sets in the west.

tony draper
7th May 2013, 20:33
Assuming your nearness to the equator means the sun would be passing directly overhead at noon,you would walk in one direction until noon then do a 180 as it passes overhead then walk back to where you started from so at noon you would be as far away from the crash site as you would get,and at sunset back where you started from,ere I think

7th May 2013, 20:34
The correct answer (but probably not the one in the simplistic marking system) is insufficient information, both answers depend on the date.

Now if they had said it was March 21st (or September 21st) the answer would probably be the easy one.

7th May 2013, 20:42
If you follow the sun all day you are not walking in a straight line
but one big curve.

tony draper
7th May 2013, 21:00
I think the equator is mentioned so for the purposes of the question we would assume the sun would be directly overhead,tiz a question being directed at sprogs and proon crumblies yer know,

7th May 2013, 21:06
somewhere near the equator
How near? :confused:

tony draper
7th May 2013, 21:10
As someone has already said insufficient data, we would need the time of year as well as the Sun bounces betwixt Capricorn and Cancer.

7th May 2013, 21:18
Would the problem be significantly different at other latitudes?

7th May 2013, 21:22
Draper. Cast your single remaining neuron back to Waddy Scorchio.

You have the answer.

7th May 2013, 21:22
At what time of the day were you farthest from the crashed aircraft?

Scene-of-the-crash time?
Or where-you're-standing-at-the-time time?

tony draper
7th May 2013, 21:23
In higher latitudes you would walk in a semi circle surely,:confused:
Good grief! is that yourelf Mr fantom?

7th May 2013, 21:25

"I think the equator is mentioned so for the purposes of the question we would assume the sun would be directly overhead,................

At midday yes, but the sun still has to rise and set on the horizon.

Not 100% sure what you are getting at.

7th May 2013, 21:27
"In higher latitudes you would walk in a semi circle surely,http://images.ibsrv.net/ibsrv/res/src:www.pprune.org/get/images/smilies/confused.gif"

I think in any latitude you walk in a semi circle.

Happy to be corrected.

tony draper
7th May 2013, 21:36
Well if you were standing on the Equator at the equinox? facing the sun as it rose,and you remained in the same spot all day it would appear to rise in a straight line at 90 degrees to the horizon pass over your noggin and set directly behind your back,ergo to follow it you would walk in a straight line toward it until noon then reverse your direction and retrace your steps?

7th May 2013, 21:42
Don't follow the sun


7th May 2013, 21:47
Don't follow the sun
So, how would you 'navigate'? (without a compass or similar device)

Wait until night-time and follow the stars, but which one(s)?

7th May 2013, 21:48

Re your last post, yes within reason but not sure the sun
follows the equator exactly.

tony draper
7th May 2013, 21:50
Well if the question means you walk all day in the direction of the rising sun rather than following the sun assuming a walking speed of two mph,you will be 24 miles from the crash site at sunset,but it does not give us a walking speed?
I am old bald and feeble minded but I'm sure the sun would follow the equator all day at the Equinoxes other tImes because we are tilted 23 1/2 degrees from the plane of the ecliptic it will be directly overhead somewhere betwixt the tropics of Capricorn and Cancer,that's why said tropics lie 23 1/2 degrees North and South of the Equator,they are the limit of how far the sun appears to drift North or South the equator.

7th May 2013, 22:05
Assuming a constant walking speed:
On the equator, at the equinox, you will walk due east until solar noon, then due west until sunset, ending up at your start point.
At any other time of year or latitude, you will walk NE or SE until solar noon, then NW or SW until sunset, finishing up north or south respectively of your start point.
In an arctic or antarctic summer, depending on your start time, you will walk a sinusoidal curve until you reach the pole, or until winter, whichever comes first.

7th May 2013, 22:24

You are at the equator at Equinox and there are no obstructions in either direction, so the day at surface level is 12hrs and the sun will move directly over you.
You will move at a constant average walking speed of 5km/hr.
Your eye height is 2m.

Radius of Earth: 6371km
Rotational speed of Earth: 1667.924km/h (15°/hr)
Rotational speed of you walking = 5 * 15/1667.924 = 0.04497°/hr

Distance of your horizon = SQRT([Your height]^2 + 2*[Radius of Earth]*[Your height]) = 5.06km
Declination to your horizon = 90 - Arctan([Radius of Earth]/[Your height]) = 0.000792°

Taking you as the stationary frame of reference, this means that as the sun moves towards you (i.e. you are walking east towards the sun) then the sun is moving towards you at 15.04497°/hr and as the sun moves away from you (i.e. you are walking west towards the sun) the the sun is moving away from you at 14.95503°/hr.

To get overhead you the sun must travel 90+declination = 90.000792°, and it must travel the same rotation to get to the horizon again.

Time taken to get overhead you = 90.000792/15.04497 = 5.98212 hrs
Time taken to reach other horizon = 90.000792/14.95503 = 6.018093 hrs

Distance moved east = 5.98212h * 5km/h = 29.9106km
Distance moved west = 6.018093h * 5km/h = 30.09047km
Total distance travelled = 60.00107km
Total movement = 30.09047 - 29.9106 = 0.179868km, or ~180m to the west.

Varying your eye height between 2m and 0 will reduce the total movement by 2mm as you reach zero, and will reduce the overall distance travelled by 53cm.

How far from the crash site have you gone?~180mAt what time of the day were you farthest from the crashed aircraft?5hrs 58mins 55.6secs after sunrise, whatever time that was.

On a similar note, this (http://what-if.xkcd.com/25/) is quite interesting.

7th May 2013, 22:26

Always somebody who has to ruin a drunken ramble with hard logic :p

7th May 2013, 22:28
Hydromet is exactly right. In numerical terms, you will walk in a semi circle on any day except an equinox where the half circumference (distance walked) is equal to 12 times your walking speed (v). From Sep-Mar, you will half-circle south; north for the other six months. You will always walk for 12 hours if we assume the Earth's orbit is circular.
Thus you will be furthest away at the end of the day, by a distance of 1 diameter of the circle, which works out to be a distance of 24 times your walking speed (in units per hour) divided by PI. For an average 2 mph, this comes out as about 15 miles away, due south or north of the start point depending on the time of year.

p.s. PTT is correct with the correction for the Earth's spin, but having taught this to 15 year olds for some time, my solution is what your kid's teacher wants.

7th May 2013, 22:30
After surviving a crash in the desert somewhere near the equator.
OK, were you navigating or sleeping at the time?
You get up at sunrise and start walking towards the sun.
Why? You'll never get to the sun. Walk towards somewhere that might have water, or maybe stay at the crash site and await the Search and Rescue effort. You did file a flight plan, right?
You continue all day walking towards the sun until sunset, then you stop.
At this point you ask yourself: why was I trying to reach the sun? It's hotter there than it is in the desert.
How far from the crash site have you gone?
Depends. How fast did you walk, how often did you rest, did you walk at a constant speed ...
At what time of the day were you farthest from the crashed aircraft?

Probably noon, but with the above questions unanswered, not sure.

In a survival situation, I don't want this person making a decision. The person is an idiot.

Note: once your plane crashes, in a desert, you are until aid arrives in a survival situation.

This nitwit is more likely to die than not.

7th May 2013, 22:34
I had made the exact same calculation as PTT, and was just about to post it, when a person from Porlock rang my doorbell and distracted me. I am mortified.

Anyway, I confirm that PTT's calculations are correct.

7th May 2013, 22:34
In which direction would you be walking at exactly noon? Can you walk "up"??

7th May 2013, 22:38
PTT, SSK are correct on the equinoxes, otherwise it's a semi-circle as I said (with a slight 'fix' for rotation).

Ascend Charlie
8th May 2013, 01:31
Insufficient information. You might be walking towards the sun, reach the Saudi border and they arrest you and drive you away to jail - that will be the furthest you get. Otherwise, you walk 2km, find a pub and stay there all day.:8

8th May 2013, 04:12
Do the Saudis have a border near to the equator then....?!

8th May 2013, 06:03
After surviving a crash in the desert somewhere near the equator.Forgive me for returning to basics but why do you not shelter from the sun under the wreckage and let the ELT do its job?

8th May 2013, 06:17
Do the Saudis have a border near to the equator then....?!

Correct !

So the answer is :More information required,there are no deserts 'somewhere near the equator'

8th May 2013, 07:13
So far there has been no mention of the coriolis effect.
I am sure you have to allow for this.

8th May 2013, 07:15
I am sure you have to allow for this. - come on, Mike - you know you are just teasing :) - at the speed of walking and 'near' the equator, it will only be felt as a slight lean one way or the other......................

8th May 2013, 07:23
BOAC, do you have to factor in which way a gentleman (ahem) dresses when calculating the lean to left or right?

8th May 2013, 07:40
....do you really need to ask? Gentlemen change with hemisphere, of course.

tony draper
8th May 2013, 07:44
Read somewhere that right handed people lost in dense jungle or featureless desert tend to walk in a clockwise circle lefties anticlockwise,hmmm, or is it the other way round,perhaps best to let a ambidextrous person lead the way.

8th May 2013, 07:58
So if our crash survivor crosses the equator several times as he walks, is he just playing with himself?

tony draper
8th May 2013, 08:08
Got lost in the outback once, about thirty miles outside of Port Pirie, I wasn't lost and knew exactly where I was and which way to go but the other six blokes disagreed,none of them were lost either and each knew exactly where he was and in which direction to head off, all of them different.
We lit a large fire and the Safari bloke found us about six hours later.

If you get lost tiz best to get lost alone.

8th May 2013, 11:05
:eek::eek: Will give PTT's solution to my lad.

Prefer Ascend Charlie's myself :ok::)

8th May 2013, 11:56
As Fox 3 said, his is likely the answer the teacher wants.

8th May 2013, 12:49
insufficient data

Is there a pub located half-a-day's walk due east of the crash site ? If so that's not only the furthest you will travel that day, but where the sun will find you when it comes up next morning. So then, continue walking due east until you reach the next pub....and so on ad. inf.

8th May 2013, 15:37
If the lad were 17, then PTT's would be the most likely 'required' answer.

find a pub and stay there all day


Ant T
16th May 2013, 12:42
I realise that the “simple” answer may be what the kid’s teacher wants, but the answers given above did not seem anywhere near right to me. The only answer that did look correct was the special case of “on the equator, at the equinox”, which does appear to me to be due east from dawn till local noon, then due west till sunset, arriving back where you started. (Presumably, this is the special case that the original question was referring to)

I was having trouble visualising any other case, but finally found this great app/webpage - SunCalc - sun position, sunlight phases, sunrise, sunset, dusk and dawn times calculator (http://www.suncalc.net/#/11.1524,-2.3201,1/2013.05.16/19:23)

I am no expert at this subject,but using that calculator to look at various cases, the only place I can see where you would walk anything close to a semi-circle would be at one of the Poles, at mid-summer, and only if you walked for just 12 hours (if you walked till dark it would be a long walk……..and incidentally, seems to me would be some kind of spiral AWAY from the Pole, not towards it as Hydromet suggested).

Some cases looked very complicated – starting at sunrise in West Africa today (May 16th) and walking till sunset, the first hour or so you slowly veer right, then virtually straight for a couple of hours, then rapidly to the left for two or three hours, then straight for another few, before finally veering right again for the last hour.

So, have I misunderstood the question, or misunderstood the explanations in the posts above, or what?

(Incidentally, I have stood right at the South Pole at mid-summer a couple of times, when I was working for British Antarctic Survey, and it was thinking about the situation there that made me think the explanations above couldn’t be quite right. It was amazing to see the sun stay exactly the same height above the horizon as it moved horizontally to the left)

Lord Spandex Masher
16th May 2013, 12:52
What if you take a pace north, or south, as you turn around?

Ant T
16th May 2013, 13:51
What if you take a pace north, or south, as you turn around?

If that question was to me, with reference to standing right at the South Pole, then the answer is that you would keep walking away from the Pole.

I think you must be picturing the sun being "Straight up" at mid-summer at the Pole - actually even at mid summer it is only low on the horizon (23.5 degrees ??? I presume)

So, as far as I can see, starting at the Pole you would just walk a big spiral away from your starting point - the first 12 hours would be approximately a semi-circle, but not precisely.

Lord Spandex Masher
16th May 2013, 14:32
Ant, no just a generally question, see if anyone considered divergence...or is it convergence...or is it something else I've forgotten the name of?!

16th May 2013, 16:24
As ever, the clue to the answer is in the question.......

You're in the desert...next to a big column of smoke from the Crash......

Whichever direction you start walking within an hour you'll get kidnapped by one whacko group or another, and end up transported hundreds of miles away, or photographed, chopped up and buried.

Take your pick.

The one place you will NOT be 24 hours later is stood near the original site.

QED.......... :p