[Guest]

Let's go back to childhood...

It's Christmas, and your parents just got you a shiny brand new bicycle! So naturally you took it for a ride around the neighborhood. But oh no! Here comes mean old kyle in middle school, and he wants your bike! He pushes you off, and rides off. He gets away, but you run after him, and eventually come upon a muddy spot in a field where he has just been on your bike. You know he is either riding south to his clubhouse, or west to his girlfriend's house to give her the bike. He is not going east or north. In the mud, you find two sets of tracks.

And

Is it possible to figure out which tracks you should follow? (specifically, can you figure out if the bikes are traveling east or west, north or south?) If so, how would you? (Bonus points if you actually do it )

EDIT:

You cannot tell which track is deeper.

Edited October 24, 2008 by Noct

[Guest]

Let's go back to childhood...

It's Christmas, and your parents just got you a shiny brand new bicycle! So naturally you took it for a ride around the neighborhood. But oh no! Here comes mean old kyle in middle school, and he wants your bike! He pushes you off, and rides off. He gets away, but you run after him, and eventually come upon a muddy spot in a field where he has just been on your bike. You know he is either riding south to his clubhouse, or west to his girlfriend's house to give her the bike. He is not going east or north. In the mud, you find two sets of tracks.

And

Is it possible to figure out which tracks you should follow? (specifically, can you figure out if the bikes are traveling east or west, north or south?) If so, how would you? (Bonus points if you actually do it )

My thought is that the front tire always leads. So if one line swings out 'wildly' from the other, then the second line (back tire) must 'correct', and eventually turn, albeit less drastically, in the same direction. The only line that seems to do that is the South-heading line. AM I even close?

[Guest]

My thought is that the front tire always leads. So if one line swings out 'wildly' from the other, then the second line (back tire) must 'correct', and eventually turn, albeit less drastically, in the same direction. The only line that seems to do that is the South-heading line. AM I even close?

are on the right track, thinking about how front and rear tires might move differently. But i should have mentioned, we know both tracks were made by bicycles. So both tracks have a front and rear tire.

Also, if it swings wildly, will that tell us if it was moving east or west/north or south? Or only which tire was the front?

[Guest]

Is it possible to figure out which tracks you should follow? (specifically, can you figure out if the bikes are traveling east or west, north or south?) If so, how would you?

the clubhouse.

The rear tire will generally take a straighter path. Also, the curvature of the front tire should be leading that of the rear, so it appears that those two sets of tracks are headed S and E.

[Guest]

I wonder...

It's mud, than you can probably choose the track that's less dry, assuming he "just" passed through the mud spot.

Since nothing related to to the other track being from the same time is posted in the OP, than I think you can do it based on the "age" of the tracks.

[Guest]

the clubhouse.

The rear tire will generally take a straighter path. Also, the curvature of the front tire should be leading that of the rear, so it appears that those two sets of tracks are headed S and E.

Spot on! Congrats!

Now: I have no idea of your experience with math, but can you think of a way to always be sure without just "eyeballing it"? In other words, a method that will give a guaranteed correct direction (except in the case of going exactly straight where the paths overlap)

[Guest]

I wonder...

It's mud, than you can probably choose the track that's less dry, assuming he "just" passed through the mud spot.

Since nothing related to to the other track being from the same time is posted in the OP, than I think you can do it based on the "age" of the tracks.

Wow! I hadn't even thought of that. I like this solution . It definitely works and fits all the conditions. (I did have a different approach, but this works just as well!)

[Guest]

Spot on! Congrats!

Now: I have no idea of your experience with math, but can you think of a way to always be sure without just "eyeballing it"? In other words, a method that will give a guaranteed correct direction (except in the case of going exactly straight where the paths overlap)

Well, I suppose you could always compute the curvature of each track d2r/ds2. If the frame is rigid, the rear tire's curvature should be a scaled version of that of the front tire, shifted by an amount roughly equal to the wheelbase. If the shift is positive, the bike was travelling in the -s direction, and vice-versa.

[Guest]

Well, I suppose you could always compute the curvature of each track d2r/ds2. If the frame is rigid, the rear tire's curvature should be a scaled version of that of the front tire, shifted by an amount roughly equal to the wheelbase. If the shift is positive, the bike was travelling in the -s direction, and vice-versa.

It's actually simpler, and could be done with a stick. Assuming all the conditions are supplied, including a constantly turning bike. It does involve the curvature though

[Guest]

Tell by the thickness of the track. If the back tire is sliding it should leave a wider crossection. Also couldn't you see a trackwall since this is mud. So at any intersection point you could tell which direction the track was going because the back tire would be the clean track.

[Guest]

Tell by the thickness of the track. If the back tire is sliding it should leave a wider crossection. Also couldn't you see a trackwall since this is mud. So at any intersection point you could tell which direction the track was going because the back tire would be the clean track.

My first thought as well, however, it will be the clean track regardless of which direction you are headed

[Guest]

It's actually simpler, and could be done with a stick. Assuming all the conditions are supplied, including a constantly turning bike. It does involve the curvature though

Hmmm. I think this works...

Place one end of the stick at a point on the front tire track, preferably where the two tracks are farthest apart, and lay the stick tangent to the rear tire track. If you cannot make any other tangents with the rear track within a distance less than the bike's wheelbase (say, 4 ft.), then that is where the rear tire was, and the bike was obviously going in the other direction.

If you are unlucky enough to find two (or more) tangent points nearby at all locations, I think you can drag the stick along and see if its other end remains tangent to the rear track at always the same point. If it does, you are dragging the stick in the direction of travel.

[Guest]

Hmmm. I think this works...

Place one end of the stick at a point on the front tire track, preferably where the two tracks are farthest apart, and lay the stick tangent to the rear tire track. If you cannot make any other tangents with the rear track within a distance less than the bike's wheelbase (say, 4 ft.), then that is where the rear tire was, and the bike was obviously going in the other direction.

If you are unlucky enough to find two (or more) tangent points nearby at all locations, I think you can drag the stick along and see if its other end remains tangent to the rear track at always the same point. If it does, you are dragging the stick in the direction of travel.

Yes this is it exactly! Congrats!

One of the tracks will always be tangent to the other at a constant distance from it. You can figure out which way the bike is going by just using a stick and seeing which direction you move it in to create this.

Edited October 25, 2008 by Noct

[Guest]

Yes this is it exactly! Congrats!

Thanks, and nice puzzle!