Flipping a Coin

Introduction

The tossing of a single coin would seem to be a simple mechanical process. Given knowledge of the coin's initial position, and velocity it would seem the outcome of a coin toss should be predictable. Why do we consider flipping a coin to be the quintessential random event?

Assumptions

&circlef;    When a tossed coin lands, the side that first contacts the ground stays facing down. The coin does not bounce off the ground or roll over once it has made contact.

&circlef;    We will ignore friction and air resistance.

&circlef;    The coin travels up and down along a straight line.

&circlef;    The coin is initially "tails".

[Graphics:Images/index_gr_1.gif]: initial position of the coin
[Graphics:Images/index_gr_2.gif]: initial velocity of the coin
[Graphics:Images/index_gr_3.gif]: angular velocity of the coin
g: gravitational constant

[Graphics:Images/index_gr_4.gif]

Model

Under the assumptions above the position of the coin at any time t after the coin is tossed is described by the function:

s(t) = -[Graphics:Images/index_gr_5.gif]g [Graphics:Images/index_gr_6.gif]+ [Graphics:Images/index_gr_7.gif]t + [Graphics:Images/index_gr_8.gif]

The coin hits the ground at the moment s(t) = 0. Using the quadratic formula

t = [Graphics:Images/index_gr_9.gif].

We are only interested in t > 0, so we are ignoring the other root of this quadratic equation.

To know which side of the coin will be facing up when the coin lands we must know what side was initially up and how many revolutions the coin traveled through before it lands. If it is initially "tails" and rotates less than a quarter of a revolution or more than three quarters of a revolution, it will land "tails" up. Otherwise it lands "heads" up.

In time t, the coin turns through [Graphics:Images/index_gr_10.gif] revolutions. Thus in mathematical notation, if

0.25 < [Graphics:Images/index_gr_11.gif] (mod 1) < 0.75

the coin lands "heads" up. Otherwise the outcome is "tails".

[Graphics:Images/index_gr_12.gif]

Now we can plot the outcomes of coin tosses for different choices of initial positions, initial velocities, and angular velocities.

Results

If the coin has a constant angular velocity of 1 and we let [Graphics:Images/index_gr_13.gif] and [Graphics:Images/index_gr_14.gif] each vary between 0 and 500, the outcomes appear to organize themselves into stripes.

[Graphics:Images/index_gr_15.gif]

[Graphics:Images/index_gr_16.gif]

[Graphics:Images/index_gr_17.gif]

If the coin has a constant initial velocity of 1 and we let [Graphics:Images/index_gr_18.gif] and [Graphics:Images/index_gr_19.gif] each vary between 0 and 500, the outcomes appear to organize themselves into a moire pattern.

[Graphics:Images/index_gr_20.gif]

[Graphics:Images/index_gr_21.gif]

[Graphics:Images/index_gr_22.gif]

If the coin has a constant initial position of 1 and we let [Graphics:Images/index_gr_23.gif] and [Graphics:Images/index_gr_24.gif] each vary between 0 and 500, the outcomes appear to organize themselves into a cell-like pattern.

[Graphics:Images/index_gr_25.gif]

[Graphics:Images/index_gr_26.gif]

[Graphics:Images/index_gr_27.gif]


Converted by Mathematica      October 20, 1999