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.

Ants speed is 1 diameter per minute = 2 radii per minute.
Turntable's speed is 1/2 revolution per minute = pi radian/minute.

Ant 1, on his second try, walks around the outside of the circumference. This is pi times the radius of the circle. Since the ants all travel at 2 radii per minute, it takes him pi/2 minutes. (~= 1.57 minutes)

Ant 2 is carried by the disc. The 1/2 revolution takes 1 minute.

Ant 3 adds his 2 radians per minute to the disc's pi radians per minute, for 2 + pi radians per minute, so he takes pi/(2+pi) minutes ~= 0.611 minutes.

Ant 4 is travelling at pi - 2 radians per minute and so takes pi/(pi-2) ~= 2.75 minutes.

For ant 5, we need to solve for the amount of rotation of the turntable makes before ant arrives and the time taken:

time = t and angle of rotation = a

a = pi * t

But the straight path on the turntable that the ant takes has length 2*sin((pi-a)/2) radii, and so his time is sin(a/2), and we need to solve

sin((pi-a)/2) = t = a/pi

A numeric solution comes out with

1.868027572707903 rad rotation in   0.5946116440568358 minutes

via

DECLARE FUNCTION asin# (x#)
DEFDBL A-Z
pi = ATN(1) * 4

a = 1

DO
 t = a / pi
 a = pi - 2 * asin(t)
 PRINT a, t
LOOP

FUNCTION asin (x)
 asin = ATN(x / SQR(1 - x * x))
END FUNCTION

 

Edited on October 21, 2009, 1:55 pm

Posted by Charlie on 2009-10-21 13:52:23