Six ants are marching in a straight line at a uniform speed along a plane when they come across something interesting: part of the ground ahead is moving. A large circular patch is slowly rotating. The path of the ants is aiming them straight at the center of this circle, so they stop right at the edge. They start arguing about what to do. Eventually the first ant decides to go for it. He sets out in a straight line toward the opposite side and keeps walking straight, but the constant rotation of the disc is such that by the time he reaches the other side he is exactly where he started! This walk took exactly one minute.

Unnerved this poor ant decides to take the long way around. (How long does it take him to get to the opposite side by walking around the circumference?)

The second observes this and decides to brave the disc. When he steps on, however he gets scared. He freezes and gets carried around until he gets to the other side and the first ant grabs him and pulls him off. (How long does this second ant take?)

The third ant decides to bite the bullet and see if he can go a little faster. He steps on then walks in the direction of the rotation until he gets to the first two. (How long does this take?)

The fourth ant is a bit of a showoff he decides to get on and walk counter to the rotation. (How long does he take?)

Ant number five has been watching carefully and thinks he can be even faster than the third. He steps on, takes a slight turn and walks in a straight line (along a chord) until he reaches the far edge. He picked the angle just right, because he meets the other ants just as he steps off. (What is the angle and how long does this walk take?)

The last ant is scared but he has his pride. He doesn't really want to cross but he doesn't want to chicken out and go around the outside either. Finally he works up his nerve and heads directly towards the rest of his party, correcting his course as he goes. (How long does this final crossing take?)
[Note: this part requires calculus.]

Reunited at last the ants continue their walk towards infinity.

I agree with all Charlie’s findings for ants 1 to 5, but am I alone in worrying about the destiny of Ant 6? I haven’t been able to do the exact integration to find its position explicitly in terms of time, but I believe that Ant 6 is stranded on the disc, still walking but now hardly moving!

However, I notice that Jer’s story ends with the ants all reunited, so where have I gone wrong? I’ve decided to outline what I’ve done in the hope that someone can find any errors.

Ant 6

Let O, C and S be the target point, centre and start point respectively. Let A (r, q) be the Ant’s polar coordinates relative to a pole O and initial line OS.
Taking the disc radius to be 1 unit gives C as (1, 0) and S as (2, 0).
Let R be the distance CA and Q be the angle SCA.
Disc rotational speed is pi rad/min (antyclockwise of course).

The velocity of A is made up of two parts: the walking speed of 2 units/min towards O and the disc speed of pi*R perpendicular to CA. Resolving these parallel and perp. to OA gives:
                        dr/dt = -2 - pi*Rsin(Q - q)     and      r dq/dt = pi*Rcos(Q - q)

Using properties of triangle OCA, these equations can be simplified so that they only involve r, q and the time t:

            dr/dt = -2 - pi*sin(q)      (1)             r dq/dt = pi(r - cos(q))           (2)

These are the simplest looking differential equations I can get, but I haven’t yet found an exact way of solving them. However, using some basic numerical analysis with the boundary condition r = 2, q = 0 when t = 0, I’ve traced the ant’s spiral path from (2, 0), passing through the axis at approx (0.36, 0) after about 0.6 mins and spiralling in towards its ‘final’ position at F: (0.771, -0.690).
It never quite gets there, but that’s the point where the disc’s velocity would be equal and opposite to the ant’s, and A tends to that point. This diagram shows the approximate position of  the spiral - the individual points are not significant.

|                                         *          *     
|                            *                                  *
|                      *                                               *
|                  *                                                         *
|              *                                                                 *
O---------------------------------C-----------------------------S--------
|          *
|        *
|       *
|        *             *  *
|            *         * F  *
|                    *   *

Supporting evidence…

(1) shows that  dr/dt = 0 when q = arcsin(-2/pi) = -0.6901 rads. This is the polar equation of a line, so whenever the spiral crosses this line it does so at right angles. This happens an infinite number of times as it approaches F.

(2) shows  that dq/dt = 0 when r = cos(q). This is the polar equation of a circle with diameter OC, so tangents from O to the spiral touch the spiral at the points where it intersects this circle. This also happens an infinite number of times.

These two conditions are met alternately as the ant spirals in towards F. At F both conditions are met, making OFC a right angled triangle. F therefore has polar coordinates (sqrt(1 - 4/pi^2), -arcsin(2/pi)).

So, how did the ant escape, and where have I made the error? I would really appreciate some help here - especially in solving those differential equations.

Posted by Harry on 2009-10-27 19:50:17