The Proof of Innocence
1 Cooperative Association for Internet Data Analysis (CAIDA), University of California, San Diego (UCSD), La Jolla, CA 92093, USA
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 that the car does not stop, if the following threeconditions 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.
arXiv:1204.0162v2 [physics.pop-ph] 20 Apr 2012
C (car) v x
S (stop sign)
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 traincomes closer, it appears 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 toanalyzing a picture 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 atthat critical moment the observer’s view is brieﬂy obstructed by another external object.
II. CONSTANT LINEAR SPEED
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. Policeoﬃcer O is located on that road at distance r0 from the intersection, |OS| = r0 . The angle between OC and OS is α.
To express α(t) in terms of r0 and x(t) we observe from triangle OCS that tan α(t) = leading to α(t) = arctan x(t) . r0 (5) x(t) , r0 (4)
Consider Fig. 1 schematically showing the geometry of the considered case, and assume for a moment that C’s linear velocity is constant in timet, v(t) ≡ v0 . (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 x(t) = v0 t. (2)
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)
Observer O visually measures not the linearspeed of C but its angular speed given by the ﬁrst derivative of angle α with respect to time t, ω(t) = dα . dt (3)
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
1 0.9 0.8 ω(t) (radians per second) ω(t) (radians per second) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −10 −5 0 t (seconds) 5 10 0 −10 −5 0 t (seconds) 5 10
a0 = 1m/s2 a0 = 3 m/s2 a0 = 10 m/s2
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.
FIG. 3: The angular velocity ω of C observed by O as a function of time t if C moves with constant linear deceleration a0 , comes to a complete stop at...
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