1. Methods for calculating a derivative in real life 2 ... for calculating a derivative in real life...
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Page 1 Applications of Calculus I October 15, 2008 1. Methods for calculating a derivative in real life 2. Building a quantum star 3. Thinking four-dimensionally Dr. Thomas J. Brueckner Department of Physics
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Transcript of 1. Methods for calculating a derivative in real life 2 ... for calculating a derivative in real life...
Page 1Applications of Calculus I October 15, 2008
1. Methods for calculating a derivative in real life
2. Building a quantum star3. Thinking four-dimensionally
Dr. Thomas J. BruecknerDepartment of Physics
Page 2Our goal for this afternoon: calculating various derivatives numerically.
• Numerical calculation means1) You do not have a formula, x(t) or p(x).2) You do have a table of values like the
exercise in test, #66, p. 231.3) You need to know the velocity:
v(t) = dx
dt
Page 3Our goal for this afternoon: calculating various derivatives numerically.
• Numerical calculation means1) You do not have a formula, x(t) or p(x).2) You do have a table of values like the
exercise in test, #66, p. 231.3) You need to know the velocity:
4) Or you need the pressure gradient: ∇p = dp
dx
Page 4Calculating various derivatives numerically.
• Because where the pressure gradient is large…
• …the wind is going to be intense!
Page 5Let’s start with the exercise on p. 231, concerning the Space Shuttle, STS-49.
Page 6Let’s start with the exercise on p. 231, concerning the Space Shuttle, STS-49.
Maneuver Time Speed Acc. (bw) Acc. (fw)Launch 0 0 ft/secBegin roll 10 185End roll 15 319Throttle 89% 20 447Throttle 67% 32 742Throttle 104% 59 1325Max pressure 62 1445SRB sep. 125 4151
Let’s do a basic calculation and use
clicker to check.
Page 71. Calculate the elapsed time to use in BW acceleration at 67% throttle maneuver.
Maneuver Time Speed Acc. (bw) Acc. (fw)Launch 0 0 ft/secBegin roll 10 185End roll 15 319Throttle 89% 20 447Throttle 67% 32 742Throttle 104% 59 1325Max pressure 62 1445SRB sep. 125 4151
A. 12 sec B. 27 sec C. something else
Page 82. Which event requires using an elapsed time of 12 sec in Acc. (fw) method?
Maneuver Time Speed Acc. (bw) Acc. (fw)Launch 0 0 ft/secBegin roll 10 185End roll 15 319Throttle 89% 20 447Throttle 67% 32 742Throttle 104% 59 1325Max pressure 62 1445SRB sep. 125 4151
A. Throttle 67%B. Throttle 104%C. Throttle 89% D. Another maneuver.
Page 9Why is the averaging method important?
Maneuver Time Speed Acc. (bw) Acc. (fw)Launch 0 0 ft/secBegin roll 10 185End roll 15 319Throttle 89% 20 447 24.6 f/s2
Throttle 67% 32 742 24.6 f/s2
Throttle 104% 59 1325Max Q 62 1445SRB sep. 125 4151
Page 10Why is averaging method important?
Acceleration models
0
5
10
15
20
25
30
35
40
45
50
0 20 40 60 80 100 120 140
time t
BW FW COMBO
Acceleration models
0
5
10
15
20
25
30
35
40
45
50
0 20 40 60 80 100 120 140
time t
BW FW COMBO
Page 11Let’s try another clicker question!
Page 12Acceleration models
0
5
10
15
20
25
30
35
40
45
50
0 20 40 60 80 100 120 1
time t
BW FW COMBO
3. At what time do the acceleration models seem to agree fairly well?
A B
C
Page 13Acceleration models
0
5
10
15
20
25
30
35
40
45
50
0 20 40 60 80 100 120 1
time t
BW FW COMBO
3. At what time do the acceleration models seem to agree fairly well?
B
C
A
Page 14
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
Quantum stars: tunneling events
1. Potential energy is like a hill.2. Quantum field responds like a ball
rolling on this hill.3. Except when it tunnels.
Page 15
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
What are the critical points to know for tunneling?
A. Global minimum energy, Vmin
B. Global maximum energy, Vmax
C. Local minimum energyD. Local maximum energyE. And the “height” of V(s) at each of these!!!
?
Page 16
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
What are the critical points to know for tunneling?
Cold quantum field
Hot quantum field
Page 17
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
What are the critical points to know for tunneling?
Hot quantum field
Max. speed
Turning pt.
Page 18
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
What are the critical points to know for tunneling?
Cold quantum field
NO
Tunnels, YES
Page 19What are the critical points to know for tunneling?
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
http://www.nature.com.ucfproxy.fcla.edu/news/2008/081007/full/news.2008.1155.html
Bunch of guys just won the Nobel prize for
figuring out this tunneling process for subnuclear particles!
Page 20
Quantum field energy function V(s)
-0.0005
-0.00025
0
0.00025
0.0005
0.00075
0.000 0.025 0.050 0.075 0.100 0.125
s
Let’s take a look at a quantum star
Page 21Let’s take a look at a quantum star.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 1 2 3 4 5 6 7 8 9
r (dimensionless)
s
Page 22Next week: we build a quantum star and maybe set it up in curved spacetime!
