Transcript Presentation

Proof of Innocence
Reproducing the Event
BY JOHN S. OCHAB, PH.D.
PROFESSOR OF PHYSICS
WITH
KENT RITCHIE
UNIVERSITY PHYSICS STUDENT
J. SARGEANT REYNOLDS COMMUNITY COLLEGE
R I C H M O N D , VA
HEADLINE NEWS
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Physicist Writes Mathematical Study to Avoid Traffic Ticket...
Buzz Blog: Physicist Uses Math to Beat Traffic Ticket
Dmitri Krioukov, Physicist, Writes Four Page Paper To Avoid ...1
The Event – Author’s Car Stops at Sign and Proceeds
Police uses Personal Vision
instead of Radar
Geometry of the Author’s Car approaching the Stop sign,
without obstructions, as eye-witnessed by policeman at
point, O.
v
X
10m  32.8ft
R0 =
10m

O
O
Author’s1 calculations of the Angular Velocity, , of his Car as
a function of Time, t, for Constant Speed through Stop sign
 Linear Speed of car at constant speed:
 letting 𝑡 = 0 is when car stops at sign.
𝑣(𝑡) = 𝑣0
(eq. 1)
 Stopping distance, x, of his car:
x(𝑡) = 𝑣0 𝑡
(eq. 2)
 Author claims policeman “visually” measures the Angular Speed, , not the Linear
Speed, v, of his car.
𝜔(𝑡) =
 From the diagram:
tan 𝜃 𝑡 =
𝑥(𝑡)
𝑅0
or
θ 𝑡 = tan−1
 Taking the time derivative on both sides of (eq. 4):
 Substituting (eq.4) into (eq.3):
𝑑𝜃 𝑡
𝑑𝑡
𝜔(𝑡) =
𝜔 𝑡 =
(eq. 3)
𝑥(𝑡)
𝑅0
𝑑 𝜃
𝑑𝑡
(eq. 4)
=
𝑑
𝑑𝑡
𝑣
𝑡𝑎𝑛−1 𝑅0 𝑡
0
𝑣0/𝑅0
2
1 + 𝑣0/𝑅0 𝑡 2
(eq. 5)
Angular Speed, , vs. Time, t, graphs of Author’s car based
on the author’s calculations (without obstructions)
Car passes through Stop sign at Constant Speed
(10 m/s  23.36 mph)
Author’s calculations of the Angular Speed, , as a function of
Time, t, of his Car “stopping” & “going” through Stop sign.
1
The stopping distance, x, of his car is:
(letting t = 0 is when car stops at sign.
x(𝑡) = 2 𝑎0 𝑡 2
(eq. 6)
Constant deceleration/acceleration of car:
𝑑𝑥 𝑡
𝑣(𝑡) = 𝑎0 𝑡 =
𝑑𝑡
(eq. 7)
Rearrange (eq 7):
𝑑𝑥 𝑡 = 𝑣(𝑡)𝑑𝑡
(eq. 8)
𝑥
Integrate both sides of (eq. 8):
𝑥 𝑡 =
𝑡
𝑑𝑥 =
0
Substitute (eq.9) into (eq.3):
(t) =
𝑑
𝑑𝑡
𝑡
𝑣 𝑡 𝑑𝑡 = 𝑎0
0
𝑡𝑎𝑛−1
0
𝑥(𝑡)
𝑅0
=
𝑑
𝑑𝑡
(eq.10)
𝑎0/𝑅0 𝑡
𝜔(𝑡) =
2
1
1 + 4 𝑎0/𝑅0 𝑡 4
1
𝑡𝑑𝑡 = 𝑎0 𝑡 2
2
𝑎0
𝑡)
2𝑅
𝑡𝑎𝑛−1 (
(eq. 9)
Graphs3 of
Distance vs. Time
and
Speed vs. Time
of a car
“stopping” and
“going” at a traffic
light
Model was based on
extensive real-life
driving data representing
general driving conditions,
involving a wide range of
speed change cycles on
different road types.
Distance
Speed
Deceleration
is not a constant.
Acceleration
is not a constant.
Angular Speed, , vs. Time, t, graphs of Author’s car based
on his calculations (without obstructions)
Car Decelerates (to full stop) and Accelerates away from the
Stop sign at various, but constant, accelerations, a0.
Angular Speed, , vs. Time, t, graphs of Author’s car based
on the author’s calculations (without obstructions)
Car passes through Stop sign at
Constant Speed
(10 m/s  23.36 mph)
Car Decelerates (to full stop) and
Accelerates away from the Stop sign
at various rates.
Desmos Graphing Calculator
https://www.desmos.com/calculator
Angular Speed when traveling at Constant Speed, v0,
through Stop sign:
𝜔(𝑡) =
𝑣0/𝑅0
1 + 𝑣0/𝑅0
2 2
𝑡
=
8𝑚/𝑠
10𝑚
8𝑚/𝑠
1 + 10𝑚
2
𝑡2
Angular Speed when Decelerating, 𝑎0, to a full stop at
Stop sign and Accelerating, 𝑎0, away from Stop sign:
𝜔(𝑡) =
𝑎0/𝑅0 𝑡
2
1
1 + 4 𝑎0/𝑅0 𝑡 4
=
1𝑚/𝑠 2
10𝑚 𝑡
1 1𝑚/𝑠 2
1 + 4 10𝑚
2
𝑡4
Our Setup – Top View
Our Setup – Side View
Measuring the Angular Speed, , and Linear Speed, v, of the
hand-pushed Black box guided by the edge of a Pasco track.
v
My hand
Top view
Laptop
x
Pasco Linear
Motion Sensor
X = 20cm
R0
r0


Ointo
= 22.7cm
Pasco Rotary Sensor
Pasco Scientific
850 Universal Interface
Demonstration of Data Acquisition
Angular Speed when traveling at Constant Speed, v,
through Stop sign.
Angular Speed when Decelerating, to a full stop at
Stop sign and Accelerating,
away from Stop sign.
Comparison of Angular Speed vs. Time graphs for Author’s
car and for Black box, both traveling at Constant Speed
through Stop sign
Author’s Car
Black box
Sample rate: 40Hz
Angular Speed versus Time graphs as before, including a
Linear Speed versus Time graph of the Black box
Author’s Car
Black box
Sample rate: 40Hz
Angular
Velocity,

Linear
Speed, v
Comparison of Angular Speed vs. Time graphs for Author’s
car and Black box moving at a faster Constant Speed through
Stop sign
Author’s Car
Black box
Sample rate: 40Hz
Angular
velocity, 
Linear
Speed, v
Comparison of Angular Speed vs. Time graphs for Author’s
car and Black box both “stopping” and “starting” at Stop
sign.
Author’s Car
Black box
Sample rate: 40Hz
Peak-to-peak time duration =
Peak-to-peak time duration = 0.575s
Comparison of Angular Speed vs. Time graphs for Author’s
car and lab box both “stopping” and “starting” at Stop sign.
Author’s Car
Black box
Sample rate: 40Hz
Peak-to-peak time duration = 0.425s
Angular Speed vs. Time and Linear Speed vs. Time graphs of
Black box “stopping” and “starting” faster at Stop sign.
Sample rate: 40Hz
Peak-to-peak time duration = 0.275s
Peak-to-peak time duration = 0.225s
Angular and Linear Velocities vs. time graphs of Black box
“stopping” and “starting” faster at Stop sign.
Sample rate: 40Hz
Peak-to-peak time duration = 0.200s
Peak-to-peak time duration = 0.175s
(My hand cannot push any faster!)
Comparison of Angular Speed vs. Time graphs for Author’s
car and Black box both “stopping” and “starting” at Stop
sign.
Author’s Results
Our Results
t =0.175s
t =0.275s
Definitions of Full and Partial Obstructions by Another Car
Lane 1
Lane 2
Suburu?
Full Obstruction,
Subtract car lengths
O
Partial Obstruction,
Add car lengths
O
Author’s Diagram Showing Full Obstruction by Another Car
when both cars are near the Stop sign.
Toyota Yaris
Lane 1
Suburu?
Lane 2
Assume author’s car was at its
maximum possible deceleration
rate:
𝑎0 = 𝑎𝑚𝑎𝑥 = 10 m/s2
 23.4 mph/s
Author’s length, l1 = 3.81m
l1/ l2 = 0.8
R0 = 10m
Other car length, l2 = 4.80m
t = time period of obstruction
= 2𝑥/𝑎0
xpartial = l1 + l2 = 8.61 m
xfull = l2 - l1 = 0.990m
O
tpartial =
tfull =
2𝑥𝑝𝑎𝑟𝑡𝑖𝑎𝑙 /𝑎0 = 1.31s
2𝑥𝑓𝑢𝑙𝑙 /𝑎0
= 0.445s
Author’s calculations for determining the time at which his
Car’s Angular Velocity goes over it’s Maximum Value (without
obstructions).
𝑑𝜔
= 𝛼(𝑡) = 0
𝑑𝑡
 To determine this time , t’, we set the derivative:
 Using standard differential rules:
𝑑𝜔
𝑑𝑡
3
= 𝛼(𝑡) =
(eq.11)
2
4
𝑎0 1− 4 𝑎0/𝑅0 𝑡
4
𝑅0 [1+1 𝑎0/𝑅 2 𝑡 4 ]2
0
4
(eq.12)
Now 𝛼(𝑡) = 0 when the numerator in (eq. 12) is set to zero.
 Filling in for R0 = 10m and letting a0 = 10m/s2 (the maximum possible deceleration of
the author’s-type car), we get:
t’=
4
4
3
𝑅0/𝑎0 =
4
4
3
10𝑚
10𝑚/𝑠2
tfull = 0.445 s < t’ <
= 1.07s
tpartial = 1.31s
Author’s Calculations of Time Duration, tb, due to obstruction by
buildings, adjacent to lane, L2, on opposite sides of the intersection
Lane 1
Lane 2
L  10m
R0 = 10m
L = Distance between buildings
tb =
2𝐿/𝑎0
 1.07s
 t’
O
Angular Velocity vs. Time Graphs of Author’s car with
obstructions from another car and from buildings on
opposite sides of the intersection.
Dotted lines show time durations for partial
obstruction, tp, and for full obstruction tf.
t’ is the time duration at amax.
Since tb  t’, we can cut out obstruction
from buildings.
Greater proof!
Determining the time at which the Black box goes over it’s
Maximum Value (without obstructions).
Sample rate: 40Hz
Peak-to-peak time duration = 0.175s
 =0
a = -0.540 m/s2
Diagram Showing Partial Obstruction of Black box by a
fictitious box when both boxes are near the Stop sign
L1 = Lane 1
80% longer
Black box length, l1 = 0.140m
t = 2𝑥/𝑎𝑚𝑎𝑥
l1/ l2 = 0.8
t’=
80% longer
xfull = l2 - l1 = 0.035m
𝑎𝑚𝑎𝑥 = − 0.540m/s2
R0 = 22.7cm
Obstruction length, l1 = 0.175m
xpartial = l1 + l2 = 0.315m
L2 = Lane 2
4
4
3
0.2270𝑚
0.540𝑚/𝑠2
= 0.697s
tpartial = 1.08s
O
tfull = 0.360s
tfull < t’ < tpartial
Comparison of  vs. graphs for Author’s car and Black box
both “stopping” and “starting” at Stop sign showing Full and
Partial Obstruction Time Intervals
t partial
Peak to peak
t = 0.175s
t full
t’
-1.2
-1.0
-0.8
-0.6
0
0.6
0.8
1.0
1.2
Conclusions
 Our experimental results seem to substantiate the author’s
claim that the policeman’s visual measurements were
delusional.
 Peaks in the Angular Speed graphs for Stop and Go
approach peak in the Angular Speed graphs for Non-stop, as
the rate of deceleration/acceleration of the car increases.
 In reality, a car could not go that fast to produce that result.
 Obstructions due to a nearby car adds to the delusion.
 Buildings’ contributions can be ignored.
 A better experimental method is needed for stopping and
starting the black box (or other object) faster, and more closely
associated with a real car.
 Suggestions for a better method?
 Good lesson plan for students?
How I Fought Traffic Tickets
using Physics
“I could not determine if the light was red. It
was beyond the 30 limit of my field of view.”
“I was not speeding due to the
Cosine Effect”.
30
http://copradar.com/preview/chapt2/ch2d1.html
Terminix
THANK YOU!
References:
[1] [ http://arxiv.org/pdf/1204.0162v2.pdf
[2]
Acceleration and deceleration models, Rahmi Akçelik and Mark Besley,Akcelik & Associates Pty Ltd,
23rd Conference of Australian Institutes of Transport Research (CAITR 2001), Monash University,
Melbourne, Australia, 10-12 December 2001 Revised: 11 July 2002
[3]
Ibid.
Extra slides follow
Comparison of  vs. graphs for Author’s car and lab box both
traveling at “constant speed” through stop sign
Author’s Car
Comparison of  vs. graphs for Author’s car and lab box
moving at a different “Constant Velocity” through stop sign
Black box
Author’s car
Count rate: 20Hz
Comparison of  vs. graphs for Author’s car and lab box both
traveling at “constant speed” through stop sign with faster
count rate.
Author’s Car
Black box
Count rate: 40Hz
Angular velocity, , versus time, t, of car “stopping” and
“starting” at stop sign.
Count rate: 20Hz
Author’s Car
Experimental
Graph
Black
box
Peak-to-peak time duration = 0.900s
Car “stopping” and “starting” at stop sign.
Count rate: 40Hz
Peak-to-peak time duration = 0.550s
Car “stopping” and “starting” at stop sign.
Experimental Graph
Count rate: 40Hz
Peak-to-peak time duration = 0.450s
Partial Blocking of Black box with Purple box that is assumed
moving at Constant Velocity parallel to Black box.
v
My hand
Top view
Laptop
x
Pasco Linear
Motion Sensor
X = 20cm
R0
r0


Ointo
= 22.7cm
Pasco Rotary Sensor
Pasco Scientific
850 Universal Interface
Comparison of  vs. graphs for Author’s car and Black box
both “stopping” and “starting” at Stop sign showing Full and
Partial Obstruction Time Intervals
t
partial
t’
t full
Peak to peak
t = 0.175s
Diagram Showing Partial Obstruction of Black box by a
fictitious box when both boxes are near the Stop sign
L1 = Lane 1
80% longer
Black box length, l1 = 0.140m
t = 2𝑥/𝑎𝑚𝑎𝑥
l1/ l2 = 0.8
t’=
80% longer
xfull = l2 - l1 = 0.035m
𝑎𝑚𝑎𝑥 = − 2.265m/s2
R0 = 22.7cm
Obstruction length, l1 = 0.175m
xpartial = l1 + l2 = 0.315m
L2 = Lane 2
4
4
3
0.2270𝑚
2.265𝑚/𝑠2
= 0.340s
tpartial = 0.527
O
tfull = 0.176
tfull < t’ < tpartial