The Proof Of Innocence

The Proof of Innocence
Dmitri Krioukov1
1
Cooperative Association for Internet Data Analysis (CAIDA),
University of California, San Diego (UCSD), La Jolla, CA 92093, USA

arXiv:1204.0162v1 [physics.pop-ph] 1 Apr 2012

We show that if a car stops at a stop sign, an observer, e.g., a police officer, located at a certain
distance perpendicular to the car trajectory, must have an illusion thatthe car does not stop, if the
following three conditions are satisfied: (1) the observer measures not the linear but angular speed
of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a
short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at
the moment when both cars are near the stop sign.

I.

C(car)

INTRODUCTION

v

It is widely known that an observer measuring the
speed of an object passing by, measures not its actual
linear velocity by the angular one. For example, if we
stay not far away from a railroad, watching a train approaching us from far away at a constant speed, we first
perceive the train not moving at all, when it is really far,
but when the train comes closer, itappears to us moving faster and faster, and when it actually passes us, its
visual speed is maximized.
This observation is the first building block of our proof
of innocence. To make this proof rigorous, we first consider the relationship between the linear and angular
speeds of an object in the toy example where the object moves at a constant linear speed. We then proceed
to analyzing apicture reflecting what really happened
in the considered case, that is, the case where the linear
speed of an object is not constant, but what is constant
instead is the deceleration and subsequent acceleration of
the object coming to a complete stop at a point located
closest to the observer on the object’s linear trajectory.
Finally, in the last section, we consider what happens
if at thatcritical moment the observer’s view is briefly
obstructed by another external object.
II.

(1)

Without loss of generality we can choose time units t such
that t = 0 corresponds to the moment when C is at S .
Then distance x is simply
(2)

Observer O visually measures not the linear speed of C
but its angular speed given by the first derivative of angle α with respect to time t,
ω (t) =dα
.
dt

x

r0

α
O (police officer)

FIG. 1: The diagram showing schematically the geometry of
the considered case. Car C moves along line L. Its current
linear speed is v , and the current distance from stop sign S
is x, |CS | = x. Another road connects to L perpendicularly
at S . Police officer O is located on that road at distance r0
from the intersection, |OS | = r0 . The anglebetween OC and
OS is α.

Consider Fig. 1 schematically showing the geometry of
the considered case, and assume for a moment that C ’s
linear velocity is constant in time t,

x(t) = v0 t.

L (lane)

To express α(t) in terms of r0 and x(t) we observe from
triangle OCS that

CONSTANT LINEAR SPEED

v (t) ≡ v0 .

S (stop sign)

(3)

tan α(t) =

x(t)
,
r0

(4)

leading toα(t) = arctan

x(t)
.
r0

(5)

Substituting the last expression into Eq. (3) and using
the standard differentiation rules there, i.e., specifically
the fact that
d
1 df
arctan f (t) =
,
dt
1 + f 2 dt

(6)

where f (t) is any function of t, but it is f (t) = v0 t/r0
here, we find that the angular speed of C that O observes

2
1

0.9

a = 1 m/s2
0

0.8

ω(t) (radiansper second)

ω(t) (radians per second)

0.8

0.6

0.4

0.2

2

a0 = 3 m/s

0.7

a0 = 10 m/s2

0.6
0.5
0.4
0.3
0.2
0.1

0
−10

−5

0
t (seconds)

5

10

FIG. 2: The angular velocity ω of C observed by O as a
function of time t if C moves at constant linear speed v0 . The
data is shown for v0 = 10 m/s = 22.36 mph and r0 = 10 m =
32.81 ft.

0
−10

−5...