Using algebra, graphical analysis, Excel, and VPython to come to the same conclusion.
David Weaver
Chandler-Gilbert Community College
david.weaver@cgcmail.maricopa.edu
Naughty 'Faster Than Light' Neutrinos a Reality?
Recent observations by the OPERA team have nothing to do with this presentation, but
it’s cool physics, isn’t it?
Multi-Modal Problem Solving
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Introduction
Model building in physics
Physicists don’t study the real world. Physicists build models of the physical world, and then
study these models in the hopes of gaining insight into how the world operates.
Therefore, what you will do throughout this course is build models, of increasing complexity,
of the real world and by closely examining these models you may gain insight into how the
world operates. Some of the early models you will examine will be obviously limited, but
keep in mind that even the most advanced physicists are merely model-builders, and the
models they study are as obviously superficial to them as the models we will study are to us.
Model building is necessary because of the overwhelming complexity of the real world. To
attempt to study a real phenomenon, with all of its many details intact, is extremely difficult.
Moreover, models often allow you to focus on the important aspects of a phenomenon,
without the distracting details.
For example, a model of reality that everyone is familiar with is the road map. Imagine trying
to drive to a strange address across town using a road map that included every driveway and
alleyway! Although these details do exist, a model that tried to encompass all of these details
would be less useful than one in which everyone’s driveway was omitted. Thus, it is possible
to omit detail, to be a poorer reflection of reality, yet to be a better, more effective, and more
useful model. A useful model for driving across town would ignore driveways but probably
include most, if not all, streets. However, if your task was to drive across the state, not only
should the driveways be omitted, but so should the vast majority of side streets; probably state
and federal highways should be the only roads on the map. Thus, a good model is closely tied
to the task at hand. What can be a very useful model for one task can be useless for another.
Thus, when we build models where the effects of friction are neglected, or the shape of an
object is ignored, it is not the case that this is a deficient model of the situation. It may well be
the case that if these details were included, some important features of the scenario would be
masked by the complexity. Simplifications made in constructing models of reality are not
always limitations to the usefulness of the model, often they are the key to building a useful
and productive model.
From Spiral Physics by Paul D’Alessandris
Multi-Modal Problem Solving
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A pole-vaulter, just before touching the cushion on which she lands after a jump, is
falling downward at a speed of 10 m/s. The pole-vaulter sinks about 2.0 m into the
cushion before stopping.
The problem begins when she hits the cushion, not
Motion Diagram
Motion
Information
while she’s falling through the air.
The top of
the cushion Event 1: The instant she
is the origin. strikes the cushion.
a
Event 2: The instant she
stops.
t1 = 0 s
t2 =
r1 = 0 m
r2 = -2.0 m
v1 = -10 m/s
v2 = 0 m/s
She stops below
the origin.
a12 =
She’s initially moving downward, which
was chosen as the negative direction.
Mathematical Analysis
v2 = v1 + a12 (t 2 - t1 )
1
r2 = r1 + v1 (t2 - t1 ) + a12 (t2 - t1 )2
2
æ mö
m
1 ç 10 ÷
-2m = 0 -10 (t2 - 0) + ç s ÷ (t2 - 0)2
s
2 t2
çè
÷ø
m
+ a12 (t 2 - 0)
s
m
10
s
a12 =
t2
0 = -10
Now substitute this expression into the
other equation:
Substitute this result back into the
original equation:
m
s
a12 =
0.4s
m/s
a12 = 25
s
10
Multi-Modal Problem Solving
æ mö
5
m
ç
÷
-2m = -10 t2 + ç s ÷ t22
s
t
çè 2 ÷ø
m
m
t 2 + 5 t2
s
s
m
-2m = -5 t2
s
-2m
t2 =
m
-5
s
t2 = 0.4s
-2m = -10
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The driver of a car traveling at 16 m/s sees a truck 20 m ahead traveling at a constant speed of 12 m/s. The
car starts without delay to accelerate at 4.0 m/s2 in an attempt to rear-end the truck. The truck driver is too
busy talking on his cellular phone to notice the car.
Motion Diagram
Car
Truck
Motion Information
Object: Car
Object: Truck
Event 1: Car begins
to accelerate.
Event 2: Collision!
Event 1: Car begins
to accelerate.
Event 2: Collision!
t1 = 0 s
t2 =
t1 = 0 s
t2 =
r1 = 0 m
r2 =
r1 = 20 m
r2 =
v1 = 16 m/s
v2 =
v1 = 12 m/s
v2 =
a12 = +4 m/s2
a12 = 0 m/s2
Mathematical Analysis
At first glance, there appear to be six unknowns in the motion table. This should concern
you since you only have four equations (the two kinematic equations applied to the car
and the same two applied to the truck). However, since the car and truck collide at event
2, t2 and r2 for the car and truck must be equal at this event. Thus, the only four variables
are t2, r2, v2Car, and v2Truck. These can be determined by the four kinematic equations.
Specifically, set the position equation for the car equal to the position equation for the
truck and solve for t2:
r2Car = r2Truck
0 + 16
m
1 m/s
m
1
(t 2 - 0) + (4
)(t 2 - 0)2 = 20m + 12 (t 2 - 0) + (0)(t 2 - 0)2
s
2
s
s
2
m
m/s 2
m
16 t 2 + 2
t 2 = 20m + 12 t 2
s
s
s
m
m/s 2
0 = 20m - 4 t 2 - 2
t2
s
s
Using the quadratic formula, t2 = 2.32 s. Plugging this back into either position equation
yields,
r2Car = 0 + 16
m
1 m/s
(2.32s - 0) + (4
)(2.32s - 0)2
s
2
s
r2Car = 47.9m
Solving the two velocity equations gives:
m
m/s
+4
(2.32s)
s
s
= 25.3m / s
m
m/s
+0
(2.32s)
s
s
= 12m / s
v2Car = 16
v2Truck = 12
v2Car
v2Truck
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A car speeds through a school zone, traveling at 20 m/s. It takes the police officer 5 s to
secure the RADAR gun and start her motorcycle. She accelerates at 5 m/s/s until she
catches up to the clueless speeder.
Motion Diagram
Motion Information
Object:
Object:
Event 1:
Event 2:
Event 1:
Event 2:
t1 =
t2 =
t1 =
t2 =
r1 =
r2 =
r1 =
r2 =
v1 =
v2 =
v1 =
v2 =
a12 =
a12 =
Mathematical Analysis
http://goo.gl/Zhesy
Multi-Modal Problem Solving
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A car speeds through a school zone, traveling at 20 m/s. It takes the police officer 5 s to
secure the RADAR gun and start her motorcycle. She accelerates at 5 m/s/s until she
catches up to the clueless speeder.
http://goo.gl/YvO0m
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t (s)
Rp (m)
Vp (m/s)
0
0
1 "=B2+C3*1"
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
t (s)
Rp (m)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Rc (m)
Vc (m/s)
20
0
0
20
0
0
20
0
0
20
0
0
20
0
0
20
0
0
20 "=D7+E8*1" "=E7+F8*1"
20
20
20
20
20
20
20
20
20
20
20
Vp (m/s)
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
Multi-Modal Problem Solving
Rc (m)
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
Vc (m/s)
0
0
0
0
0
0
5
15
30
50
75
105
140
180
225
275
330
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Ac (m/s/s)
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
Ac (m/s/s)
0
0
0
0
0
0
5
10
15
20
25
30
35
40
45
50
55
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
7
t
Vp
Ac
(s) Rp (m)
(m/s)
Rc (m)
Vc (m/s)
(m/s/s)
0
0
20
0
0
1 "=B2+C3*1"
20
0
0
2
20
0
0
3
20
0
0
4
20
0
0
5
20
0
0
6
20 "=D7+E8*1" "=E7+(F7+F8)/2*1"
7
20
8
20
9
20
10
20
11
20
12
20
13
20
14
20
15
20
16
20
17
20
t (s)
Rp (m)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Vp (m/s)
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
Multi-Modal Problem Solving
Rc (m)
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
0
0
0
0
0
0
2.5
10
22.5
40
62.5
90
122.5
160
202.5
250
302.5
360
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Vc (m/s)
0
0
0
0
0
0
5
10
15
20
25
30
35
40
45
50
55
60
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
Ac (m/s/s)
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
8
## Cop N Perp by David Weaver
## BIG help from Danny Caballero of Georgia Tech
## Perp passes a cop in a school zone, traveling 20 m/s
## Cop takes off 5 s later, accelerating at 5 m/s/s
from visual import * ## includes the visual module
scene = display(forward=(0,-.7,-1)) # Tilts the camera for more of a bird's eye view
scene.fullscreen = True # Expands the window to fill your screen
## Objects in scene
ground = box(pos=vector(0,-20,0), axis=(350,0,0), width=50, color=color.green)
perp = sphere(pos=vector(-175,2,0), radius=20, color=color.red)
cop = sphere(pos=vector(-175,2,10), radius=20, color=color.blue)
## Initial conditions
perp.v = vector(20,0,0) ## Perp's velocity is 20 m/s in x-direction
cop.v = vector(0,0,0) ## Cop starts from rest.
## Trails to track ball's path (not needed, but kinda cool)
perptrail = curve(color=perp.color)
coptrail = curve(color=cop.color)
## Assign acceleration
a=vector(5,0,0) ## Cop accelerates at 5 m/s/s
## Start the clock
dt = 0.001 ## This is the time step
t=0
while t < 16.5: ## I got this value from the Excel spreadsheet
rate(3000) ## Adjusts how fast the simulation runs.
perp.pos = perp.pos + perp.v*dt ##Xf = Xi + V*t
if t>5: ## Cop takes 5 s to get started
cop.v = cop.v+a*dt ## Vf = Vi + a*t
cop.pos = cop.pos + cop.v*dt ## Xf = Xi + V*t
## Update the trail to track position
perptrail.append(perp.pos)
coptrail.append(cop.pos)
t=t+dt ## Increments time
## Range & velocity calculations (all balls starting from (-175,2,0)) change if you need to
R1 = perp.pos.x - (-175)
R2 = cop.pos.x - (-175)
V1 = perp.v
V2 = cop.v
print ("Time", t, "s")
print ("Perp's range:", R1, "m")
print ("Cop's range:", R2, "m")
print ("Perp's velocity", V1, "m/s")
print ("Cop's velocity", V2, "m/s")
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