Take a hike!
 

Two people go for a walk. At the end of the walk they are back where they started. They find that one of them's head has travelled approximately 30 feet further than the other's. They weren't making any unusual movements with their heads. How is this possible, show the math.

Fusion

They walked around Green Lake. Whoever was walking on the outside obviously travelled (and their head too) father.

Nice try, no cigar. Notice the specific distance of 30 feet.

Trif is basically right. But it wasn't such a long walk. I think that it was a pair of cosy lovers who made two loops around the gazebo. If it was only one loop around the park (lake, city, whatever) , they would have probably been jogging. Otherwise, I cannot explain why they would have been nearly 5 feet apart. In a normal lap around the parade ground, the first squad travels about 21 feet less than the second squad and 42 feet less than the third squad.

Maybe it would help if I said that it doesn't matter if the two walkers walk side by side or one in front of the other. Think 3 dimensions.

Well, it can't be that one person is almost five feet taller than the other and that they walked around the Earth at the equator or along some other great circle route, so that the circumference of the circle travelled by one head was 2*pi*(almost 5 feet) = approximately 30 feet longer than the circumference of the circle travelled by the other head, because there's no way to walk around the Earth -- no matter which route you try, you encounter an ocean somewhere, and even if the Bering Strait is frozen over (someone _did_ try to drive across the Bering Strait over the ice cover in winter a few years ago -- I saw the show on PBS), there's no way to similarly cross the Atlantic because even in winter there's open water (between the icebergs) between Greenland and Iceland and Norway, and even if you walked farther north on the Arctic Ocean ice cover [which, by the way, is being destroyed by global warming, so if you want to walk on the Arctic Ocean you'd better not put it off much longer] to get from North America to Europe you can't really walk "around" the Earth in a great-circle type of path, so I guess it must be that these people are in the future, dressed in spacesuits, and walking around the Moon or an asteroid, where there would still be that approximatley 30-foot difference in the circumferences of the circles travelled by their two heads. But that short guy is going to be taking a lot more steps than the tall guy.

Or else there were stilts involved, with a five-foot difference in height of stilts between one guy and the other, and if the stilts were long enough theoretically they could wade across oceans so maybe that's it -- the two people are on stilts, and both pairs of stilts are long enough to enable them to walk across oceans, but one pair of stilts is almost five feet taller than the other, so that after they circumperambulate at the equator their heads have travelled approximately 30 feet different distances. But the guy on the shorter stilts may still have to take a few more steps than the guy on the taller stilts.

Maybe one of the two was limbo-ing

A cigar for cheesehead, would there be any difference if they were walking around a basketball? Work out the math if one is about 7 feet tall and the other is about 2.5 feet tall.

I should probably have said that one's head travelled about 20 feet farther. Its a little more believable that one person is 6 feet tall and the other is 3 feet.

Fusion

If you reduce the difference to ~20 ft, it also makes more sense in terms of two people walking side by side.

I still go with Trif's solution. All in all, I rate your variation "unacceptable".

I protest that your "single file" solution is totally unrealistic because it is not possible to "walk" around the center of the earth.

Here are some alternate explanations for the "single" file idea. (But I don't really like any of them)

Perhaps they walked around a small asteroid. (But neither the very tall or very short person would be able to get accepted for such a trip.)

Perhaps they simply walked onto an aeroplane and flew around the world a number of times. (One head is further from the seat than the other. It takes multiple laps to make up for the smaller separation)

Perhaps one simply got a little ahead of the other and turned back for 15 ft. (10 if you use ~20). You cannot travel a negative distance and the "turning around" is not a movement of the head, per se, but rather the body.

[quote="Wackerbarth"]If you reduce the difference to ~20 ft, it also makes more sense in terms of two people walking side by side.[/quote]
Are you saying that the importance of the sensibility of walking side-by-side versus some other relative formation exceeds the importance of the sensibility of circumperambulating the Earth, oceans and all? :surprised:

[quote]I still go with Trif's solution.[/quote]
So ... as the two walked, they stayed almost five feet apart, with the two of them remaining aligned with the center of Green Lake?

Well, that does seem more feasible than circumperambulating the Earth.

[quote]All in all, I rate your variation "unacceptable".[/quote]
Ummm ... which variation is his variation? Ocean-going stilts?

I don't see where Fusion_power said exactly which part of my discussion he agreed with!

[quote]I protest that your "single file" solution[/quote]
What "single file" solution is that?

[quote] is totally unrealistic because it is not possible to "walk" around the center of the earth.[/quote]
It's lots easier than walking [u]through[/u] the center of the Earth. Or, for that matter, around the center of the [u]Moon[/u].

[i](I guess by now you've noticed that I'm "on a roll" this afternoon. ... Saayyy, if two ants walk around a dinner roll, and one ant's head travels ~20 millimeters farther ...)[/i]

[quote]Here are some alternate explanations for the "single" file idea. (But I don't really like any of them)[/quote]
Perhaps some simple modification to each explanation will make it more likeable. :)

[quote]Perhaps they walked around a small asteroid. (But neither the very tall or very short person would be able to get accepted for such a trip.)[/quote]
Or else ... the persons are the same height, but they pack different-height stilts along with their spacesuits. Ta-Da!!!

[quote]Perhaps they simply walked onto an aeroplane and flew around the world a number of times. (One head is further from the seat than the other. It takes multiple laps to make up for the smaller separation)[/quote]
Or else ... they just walked to the airplane and back, without ever actually boarding the plane, enough million times to equal going around the Earth once, and when you add up the teeny-tiny differences in distance that their heads travelled on each trip to the plane due to the teeny-tiny amount of Earth's curvature that applies to that distance to the plane and back, you find that the tall guy's head travelled ~30 feet farther.

[quote]Perhaps one simply got a little ahead of the other and turned back for 15 ft. (10 if you use ~20). You cannot travel a negative distance and the "turning around" is not a movement of the head, per se, but rather the body.[/quote]
... which is such a sensible solution that it needs no amendment.

[quote="cheesehead"][quote="Wackerbarth"]If you reduce the difference to ~20 ft, it also makes more sense in terms of two people walking side by side.[/quote]
Are you saying that the importance of the sensibility of walking side-by-side versus some other relative formation exceeds the importance of the sensibility of circumperambulating the Earth, oceans and all? :surprised:
[/quote]
No, I'm saying that it is easier for either of us to justify 20 ft than it is to justify 30 ft.
[quote="cheesehead"]
So ... as the two walked, they stayed almost five feet apart, with the two of them remaining aligned with the center of Green Lake?

Well, that does seem more feasible than circumperambulating the Earth.
[/quote]
They need not remain aligned. The only requirement is that the radial distance between their paths maintains the appropriate average separation.

[quote="cheesehead"] Ummm ... which variation is his variation? Ocean-going stilts?

I don't see where Fusion_power said exactly which part of my discussion he agreed with!

[/quote]

Any variation that allows "single file".

[quote="cheesehead"] [quote="Wackerbarth"] it is not possible to "walk" around the center of the earth.[/quote]
It's lots easier than walking [u]through[/u] the center of the Earth.
[/quote]
"Through" doesn't solve the problem. You have to go "around" something in such a manner that the average differential of the radius from that point is maintained.

[quote="cheesehead"]
Or, for that matter, around the center of the [u]Moon[/u].
[/quote]
Sorry, I already claimed that one with an asteroid. :D

(I think that we have beaten up Fusion_Power well enough. Let him lick his wounds in peace)