This document provides a mathematical proof of innocence for a driver accused of not stopping at a stop sign. The summary is:
1. An observer measuring the angular speed rather than linear speed of an object can perceive it as not stopping if the object decelerates and accelerates quickly near the observer and their view is briefly obstructed.
2. The document analyzes the relationship between linear and angular speed and shows that a car decelerating and accelerating rapidly near an observer could appear to not stop.
3. It applies this analysis to a scenario where a driver's view of an accused car was briefly obstructed when the car was near a stop sign, which could have caused the observer to incorrectly perceive
The Proof of Innocence
Dmitri Krioukov 1
1
Cooperative Association for Internet Data Analysis (CAIDA),
University of California, San Diego (UCSD), La Jolla, CA 92093, USA
Convex Hull - Chan's Algorithm O(n log h) - Presentation by Yitian Huang and ...Amrinder Arora
Chan's Algorithm for Convex Hull Problem. Output Sensitive Algorithm. Takes O(n log h) time. Presentation for the final project in CS 6212/Spring/Arora.
The Proof of Innocence
Dmitri Krioukov 1
1
Cooperative Association for Internet Data Analysis (CAIDA),
University of California, San Diego (UCSD), La Jolla, CA 92093, USA
Convex Hull - Chan's Algorithm O(n log h) - Presentation by Yitian Huang and ...Amrinder Arora
Chan's Algorithm for Convex Hull Problem. Output Sensitive Algorithm. Takes O(n log h) time. Presentation for the final project in CS 6212/Spring/Arora.
This is a lecture on the hydraulics of gradually varied flow in open channels. It shows the profiles common in the open channels and some numerical examples using numerical integration.
Θεωρία Μαθηματικών κατεύθυνσης ... η επιμέλεια έγινε από τους συναδέλφους Μπάμπη Στεργίου και Παπαμικρούλη Δημήτρη. Το υλικό αναρτήθηκε στην ιστοσελίδα http://lisari.blogspot.com/ και εμείς απλά το γνωστοποιούμε σε περισσότερο κόσμο.
This is a lecture on the hydraulics of gradually varied flow in open channels. It shows the profiles common in the open channels and some numerical examples using numerical integration.
Θεωρία Μαθηματικών κατεύθυνσης ... η επιμέλεια έγινε από τους συναδέλφους Μπάμπη Στεργίου και Παπαμικρούλη Δημήτρη. Το υλικό αναρτήθηκε στην ιστοσελίδα http://lisari.blogspot.com/ και εμείς απλά το γνωστοποιούμε σε περισσότερο κόσμο.
Research on The Control of Joint Robot TrajectoryIJRESJOURNAL
ABSTRACT: This paper relates to a Robot that belongs to the category of Joint Robot.In the article,we analyze the path planning and control system of the robot,specifically speaking,it involves the interpolation of the robot trajectory, the analysis of the inverse kinematics, the introduction of the method to reduce the trajectory error, the optimization of the trajectory and in the end, the corresponding control system is designed according to the relevant parameters. This research project first introduces the importance of the robot, and then analyzes the whole process of the robot from the grasping pin, the screw to they are delivered to the designated position,finally, the process is introduced in detail, and the simulation result is displayed.
I am Martina J. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Maryland. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
Introduction to differential calculus. This slide deals with the concept of differential calculus. An explanation is given that how derivative of a function can be calculated graphically and algebraically
Τα Μαθηματικά και η σχέση τους με την Αρχιτεκτονική σχεδίασηΡεβέκα Θεοδωροπούλου
Το άρθρο αυτό δημοσιεύτηκε στο περιοδικό Ευκλείδης γ', το οποίο εκδίδει η Ελληνική Μαθηματική Εταιρεία κάθε έξι μήνες. Το συγκεκριμένο συμπεριλαμβάνεται στο τεύχος 89, περιόδου Ιουλίου - Δεκεμβρίου 2018.
Υπάρχει κάποιος μαθηματικός τύπος που να περιγράφει το σασπένς; Το σασπένς προκαλείται από την κατάλληλη ανάμιξη του χρόνου με την προσμονή και την αβεβαιότητα, λένε οι καθηγητές οικονομικών Emir Kamenica και Alexander Frankel από το Πανεπιστήμιο του Chicago, που μελέτησαν τη δομή του σασπένς στο πλαίσιο της έρευνάς τους για τη σχέση του σασπένς με την οικονομία. Πηγή : http://thalesandfriends.org/
Η παρουσίαση αυτή έγινε από εμένα για προσωπικούς λόγους και κοινοποιείται σε περίπτωση που και κάποιος άλλος θα ήθελε να μάθει μερικά βασικά πράγματα.
εταιρική εικόνα ως σύνθετη διαδικασία, απόδειξης και τοποθέτησης διαμέσου μία...Ρεβέκα Θεοδωροπούλου
Η εργασία αυτή εκπονήθηκε από εμένα και μόνο και κάθε εμφάνιση ενός κομματιού ή ολόκληρου του κειμένου είναι αντιγραφή !!!
εταιρική εικόνα ως σύνθετη διαδικασία, απόδειξης και τοποθέτησης διαμέσου μίας μελέτης περίπτωσης, συγκριτική μελέτη
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Home assignment II on Spectroscopy 2024 Answers.pdf
Η απόδειξη της αθωότητας
1. 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:1204.0162v2 [physics.pop-ph] 20 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 that the 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, 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
to analyzing 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 at that critical 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 angle between 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 to
α(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. 2
1
0.9
a = 1 m/s2
0
0.8
ω(t) (radians per 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
0
t (seconds)
5
10
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 S at time t = 0, and then moves
with the same constant linear acceleration a0 . The data are
shown for r0 = 10 m.
as a function of time t is
but by the definition of velocity,
v0 /r0
ω(t) =
1+
v0
r0
2
.
(7)
t2
v(t) =
This function is shown in Fig. 2. It confirms and quantifies the observation discussed in the previous section,
that at O, the visual angular speed of C moving at a
constant linear speed is not constant. It is the higher,
the closer C to O, and it goes over a sharp maximum at
t = 0 when C is at the closest point S to O on its linear
trajectory L.
CONSTANT LINEAR DECELERATION
AND ACCELERATION
In this section we consider the situation closely mimicking what actually happened in the considered case.
Specifically, C, instead of moving at constant linear
speed v0 , first decelerates at constant deceleration a0 ,
then comes to a complete stop at S, and finally accelerates with the same constant acceleration a0 .
In this case, distance x(t) is no longer given by Eq. (2).
It is instead
x(t) =
1 2
a0 t .
2
(8)
If this expression does not look familiar, it can be easily
derived. Indeed, with constant deceleration/acceleration,
the velocity is
v(t) = a0 t,
(9)
(10)
so that
dx = v(t) dt.
(11)
Integrating this equation we obtain
x
x(t) =
t
dx =
0
III.
dx
,
dt
t
v(t) dt = a0
0
t dt =
0
1 2
a0 t . (12)
2
Substituting the last expression into Eq. (5) and then differentiating according to Eq. (3) using the rule in Eq. (6)
with f (t) = a0 t2 /(2r0 ), we obtain the angular velocity of
C that O observes
a0
r0 t
ω(t) =
1+
1
4
a0
r0
2
.
(13)
t4
This function is shown in Fig. 3 for different values
of a0 . In contrast to Fig. 2, we observe that the angular
velocity of C drops to zero at t = 0, which is expected
because C comes to a complete stop at S at this time.
However, we also observe that the higher the deceleration/acceleration a0 , the more similar the curves in Fig. 3
become to the curve in Fig. 2. In fact, the blue curve in
Fig. 3 is quite similar to the one in Fig. 2, except the narrow region between the two peaks in Fig. 3, where the
angular velocity quickly drops down to zero, and then
quickly rises up again to the second maximum.
3. 3
C1 (car #1)
S (stop sign)
L1 (lane #1)
tp = 1.31 s,
tf = 0.45 s.
L2 (lane #2)
C2 (car #2)
l1
l1 = 150 in
C2 ≈ Subaru Outback
l2
(15)
(16)
The full durations of the partial and full obstructions are
then just double these times.
Next, we are interested in time t at which the angular
speed of C1 observed by O without any obstructions goes
over its maxima, as in Fig. 3. The easiest way to find t
is to recall that the value of the first derivative of the
angular speed at t is zero,
C1 = Toyota Yaris
r0
we obtain
l2 = 189 in
l2 − l1 = 39 in ≈ 1 m
dω
= ω(t ) = 0.
˙
dt
(17)
To find ω(t) we just differentiate Eq. (13) using the stan˙
dard differentiation rules, which yield
O (police officer)
FIG. 4: The diagram showing schematically the brief obstruction of view that happened in the considered case. The O’s
observations of car C1 moving in lane L1 are briefly obstructed
by another car C2 moving in lane L2 when both cars are near
stop sign S. The region shaded by the grey color is the area
of poor visibility for O.
IV.
a0
ω(t) = 4
˙
r0
t=
2x
,
a0
(14)
1+
a0
r0
3
4
1
4
a0
r0
2
2
t4
2.
(18)
t4
This function is zero only when the numerator is zero, so
that the root of Eq. (17) is
t =
BRIEF OBSTRUCTION OF VIEW AROUND
t = 0.
Finally, we consider what happens if the O’s observations are briefly obstructed by an external object, i.e.,
another car, see Fig. 4 for the diagram depicting the
considered situation. The author/defendant (D.K.) was
driving Toyota Yaris (car C1 in the diagram), which is
one of the shortest cars avaialable on the market. Its
lengths is l1 = 150 in. (Perhaps only the Smart Cars are
shorter?) The exact model of the other car (C2 ) is unknown, but it was similar in length to Subaru Outback,
whose exact length is l2 = 189 in.
To estimate times tp and tf at which the partial and,
respectively, full obstructions of view of C1 by C2 began and ended, we must use Eq. (8) substituting there
xp = l2 + l1 = 8.16 m, and xf = l2 − l1 = 0.99 m, respectively. To use Eq. (8) we have to know C1 ’s deceleration/acceleration a0 . Unfortunately, it is difficult to measure deceleration or acceleration without special tools,
but we can roughly estimate it as follows. D.K. was badly
sick with cold on that day. In fact, he was sneezing while
approaching the stop sign. As a result he involuntary
pushed the brakes very hard. Therefore we can assume
that the deceleration was close to maximum possible for
a car, which is of the order of 10 m/s2 = 22.36 mph/s.
We will thus use a0 = 10 m/s2 . Substituting these values
of a0 , xp , and xf into Eq. (8) inverted for t,
1−
4
4
3
r0
.
a0
(19)
Substituting the values of a0 = 10 m/s2 and r0 = 10 m
in this expression, we obtain
t = 1.07 s.
(20)
We thus conclude that time t lies between tf and tp ,
tf < t < tp ,
(21)
and that differences between all these times is actually
quite small, compare Eqs. (15,16,20).
These findings mean that the angular speed of C1 as
observed by O went over its maxima when the O’s view
of C1 was partially obstructed by C2 , and very close in
time to the full obstruction. In lack of complete information, O interpolated the available data, i.e., the data for
times t > t ≈ tf ≈ tp , using the simplest and physiologically explainable linear interpolation, i.e., by connecting
the boundaries of available data by a linear function. The
result of this interpolation is shown by the dashed curve
in Fig. 5. It is remarkably similar to the curve showing the angular speed of a hypothetical object moving at
constant speed v0 = 8 m/s ≈ 18 mph.
V.
CONCLUSION
In summary, police officer O made a mistake, confusing
the real spacetime trajectory of car C1 —which moved
at approximately constant linear deceleration, came to a
4. 4
0.9
0.9
C1’s speed
C ’s speed
1
O’s perception
v = 8 m/s
0.8
O’s perception
v = 8 m/s
0.8
0
0
0.7
ω(t) (radians per second)
ω(t) (radians per second)
0.7
0.6
0.5
0.4
0.3
tp
0.2
0.6
0.5
0.4
0.3
0.2
t′
0.1
0
−10
0.1
tf
−8
−6
−4
−2
0
2
t (seconds)
4
6
8
10
0
−10
−8
−6
−4
−2
0
2
t (seconds)
4
6
8
10
FIG. 5: The real angular speed of C1 is shown by the blue solid
curve. The O’s interpolation is the dashed red curve. This
curve is remarkably similar to the red solid curve, showing
the angular speed of a hypothetical object moving at constant
linear speed v0 = 8 m/s = 17.90 mph.
FIG. 6: The same data as in Fig. 5 shown for t ∈ [−tb , tb ],
where tb = 1.41 s.
complete stop at the stop sign, and then started moving
again with the same acceleration, the blue solid line in
Fig. 5—for a trajectory of a hypothetical object moving
at approximately constant linear speed without stopping
at the stop sign, the red solid line in the same figure.
However, this mistake is fully justified, and it was made
possible by a combination of the following three factors:
Contrary to common belief, the problem is not that
Yaris cannot accelerate that fast. According to the official Toyota specifications, Yaris accelerates to 100 km/h
2
in 15.7 s, which translates to 1.77 m/s . However, this
is the average acceleration, which is not constant. It is
well known that most cars accelerate much faster at low
speeds than at high speeds, so that the assumption that
2
acceleration a0 was about 10 m/s was not unjustified.
This problem of what the exact value of a0 was, becomes actually irrelevant in view of that neither Yaris nor
any other car could decelerate or accelerate that fast for
that long, which the author recognized soon after arXival. Indeed, the linear speed of C1 would be too high
at t = ±10 s in that case. The deceleration/accelaration
2
a0 ∼ 10 m/s could thus last for only 1-2 seconds.
The question of how the data shown in Fig. 5 would
change (presumably not much) if we take into account
non-constant a for the whole range of t ∈ [−10, 10], is also
irrelevant in view of an additional circumstance brought
up by the judge. Both southeast and southwest corners
of the intersection in Fig. 4 are occupied by buildings,
limiting the view from O to about lb = 10 m from S
along L1 . Substituting this lb instead of x in Eq. (14),
we obtain tb = 1.41 s, which is the time of appearance
and disappearance of C1 from O’s view obstructed by the
buildings. Therefore instead of Fig. 5, we have Fig. 6,
obtained from Fig. 5 by cutting off all the data outside
the range t ∈ [−tb , tb ]. Clearly, the same conclusions
hold, even become stronger.
1. O was not measuring the linear speed of C1 by any
special devices; instead, he was estimating the visual angular speed of C1 ;
2. the linear deceleration and acceleration of C1 were
relatively high; and
3. the O’s view of C1 was briefly obstructed by another car C2 around time t = 0.
As a result of this unfortunate coincidence, the O’s perception of reality did not properly reflect reality.
Appendix A: Two common questions
1.
Is the stop sign fine that high in California?
The answer is no. The author did not really know what
the fine was since he was not fined. The fine, plus the
traffic school (which one wants to take to avoid points
on his driving record), is $287. Therefore the abstract
should have read $300, instead of $400.
2.
Are there any flaws in the argument?