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review derivation: speed of Earth around Sun

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Notes for this derivation:
from \cite{1999_Tipler_Llewellyn}, page 9

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 declare initial expr
  1. 5426308937; locally 1131405:
    \(v = \frac{d}{t}\)
    \(pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}\)
no validation is available for declarations 5426308937:
5426308937:
2 change variable X to Y
  1. 5426308937; locally 1131405:
    \(v = \frac{d}{t}\)
    \(pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}\)
  1. 1277713901:
    \(d\)
    \(pdg_{1943}\)
  2. 7476820482:
    \(C\)
    \(pdg_{3034}\)
  1. 6946088325; locally 7360652:
    \(v = \frac{C}{t}\)
    \(pdg_{1357} = \frac{pdg_{3034}}{pdg_{1467}}\)
valid 5426308937:
6946088325:
5426308937:
6946088325:
3 declare initial expr
  1. 6785303857; locally 7959026:
    \(C = 2 \pi r\)
    \(pdg_{3034} = 2 pdg_{2530} pdg_{3141}\)
no validation is available for declarations 6785303857:
6785303857:
circumference of a circle
4 change two variables in expr
  1. 6785303857; locally 7959026:
    \(C = 2 \pi r\)
    \(pdg_{3034} = 2 pdg_{2530} pdg_{3141}\)
  1. 4057686137:
    \(C\)
    \(pdg_{3034}\)
  2. 6239815585:
    \(C_{\rm Earth\ orbit}\)
    \(pdg_{1534}\)
  3. 2346150725:
    \(r\)
    \(pdg_{2530}\)
  4. 4202292449:
    \(r_{\rm Earth\ orbit}\)
    \(pdg_{6081}\)
  1. 6348260313; locally 5753220:
    \(C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}\)
    \(pdg_{1534} = 2 pdg_{3141} pdg_{6081}\)
valid 6785303857:
6348260313:
6785303857:
6348260313:
5 change three variables in expr
  1. 6946088325; locally 7360652:
    \(v = \frac{C}{t}\)
    \(pdg_{1357} = \frac{pdg_{3034}}{pdg_{1467}}\)
  1. 4057686137:
    \(C\)
    \(pdg_{3034}\)
  2. 7708501762:
    \(C_{\rm Earth\ orbit}\)
    \(pdg_{1534}\)
  3. 9753878784:
    \(v\)
    \(pdg_{1357}\)
  4. 9601500174:
    \(v_{\rm Earth\ orbit}\)
    \(pdg_{7427}\)
  5. 8135396036:
    \(t\)
    \(pdg_{1467}\)
  6. 4470433702:
    \(t_{\rm Earth\ orbit}\)
    \(pdg_{5344}\)
  1. 3046191961; locally 5320197:
    \(v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}\)
    \(pdg_{7427} = \frac{pdg_{1534}}{pdg_{5344}}\)
valid 6946088325:
3046191961:
6946088325:
3046191961:
6 substitute LHS of expr 1 into expr 2
  1. 6348260313; locally 5753220:
    \(C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}\)
    \(pdg_{1534} = 2 pdg_{3141} pdg_{6081}\)
  2. 3046191961; locally 5320197:
    \(v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}\)
    \(pdg_{7427} = \frac{pdg_{1534}}{pdg_{5344}}\)
  1. 3080027960; locally 9129246:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}\)
    \(pdg_{7427} = \frac{2 pdg_{3141} pdg_{6081}}{pdg_{5344}}\)
valid 6348260313:
3046191961:
3080027960:
6348260313:
3046191961:
3080027960:
7 declare assumption
  1. 7175416299; locally 9494155:
    \(t_{\rm Earth\ orbit} = 1 {\rm year}\)
    \(pdg_{5344} = 1\)
no validation is available for declarations 7175416299:
7175416299:
8 multiply RHS by unity
  1. 7175416299; locally 9494155:
    \(t_{\rm Earth\ orbit} = 1 {\rm year}\)
    \(pdg_{5344} = 1\)
  1. 3219318145:
    \(\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}\)
    \(365\)
  1. 8721295221; locally 9417128:
    \(t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}\)
    \(pdg_{5344} = 3\)
feed diff is 364 LHS diff is 0 RHS diff is 362 7175416299:
8721295221:
7175416299:
8721295221:
9 substitute LHS of expr 1 into expr 2
  1. 8721295221; locally 9417128:
    \(t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}\)
    \(pdg_{5344} = 3\)
  2. 3080027960; locally 9129246:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}\)
    \(pdg_{7427} = \frac{2 pdg_{3141} pdg_{6081}}{pdg_{5344}}\)
  1. 4593428198; locally 1441436:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 pdg_{6081} \pi\)
LHS diff is 0 RHS diff is pdg6081*(0.666666666666667*pdg3141 - 0.632911392405063*pi) 8721295221:
3080027960:
4593428198:
8721295221:
3080027960:
4593428198:
10 declare assumption
  1. 3472836147; locally 4133484:
    \(r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}\)
    \(pdg_{6081} = 1.496\)
no validation is available for declarations 3472836147:
3472836147:
11 substitute LHS of expr 1 into expr 2
  1. 3472836147; locally 4133484:
    \(r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}\)
    \(pdg_{6081} = 1.496\)
  2. 4593428198; locally 1441436:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 pdg_{6081} \pi\)
  1. 6998364753; locally 8698819:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 \pi\)
LHS diff is 0 RHS diff is 0.313924050632911*pi 3472836147:
4593428198:
6998364753:
3472836147:
4593428198:
6998364753:
12 simplify
  1. 6998364753; locally 8698819:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 \pi\)
  1. 4180845508; locally 1001745:
    \(v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}\)
    \(pdg_{7427} = 29.8\)
LHS diff is 0 RHS diff is 0.632911392405063*pi - 29.8 6998364753:
4180845508: error for dim with 4180845508
6998364753:
4180845508: N/A
13 declare final expr
  1. 4180845508; locally 1001745:
    \(v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}\)
    \(pdg_{7427} = 29.8\)
no validation is available for declarations 4180845508: error for dim with 4180845508
4180845508: N/A
Physics Derivation Graph: Steps for speed of Earth around Sun

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1534 variable C_{\rm Earth\ orbit}
\(C_{\rm Earth\ orbit}\)
real
  • length: 1
circumference of Earth's orbit 4
7427 variable v_{\rm Earth\ orbit}
\(v_{\rm Earth\ orbit}\)
real
  • length: 1
  • time: -1
velocity of Earth's orbit around sun
  • str_note
6
1467 variable t
\(t\)
['real']
  • time: 1
time 115
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 80
5344 variable t_{\rm Earth\ orbit}
\(t_{\rm Earth\ orbit}\)
real
  • time: 1
time of Earth's orbit around sun
  • str_note
5
6081 constant r_{\rm Earth\ orbit}
\(r_{\rm Earth\ orbit}\)
['real']
  • length: 1
radius of Earth's orbit 1.496*10^8   km
5
3034 variable C
\(C\)
['real']
  • length: 1
circumference 5
1943 variable d
\(d\)
['real']
  • length: 1
displacement 25
2530 variable r
\(r\)
['real']
  • length: 1
radius 34
3141 constant \pi
\(\pi\)
['real']
pi 3.1415   dimensionless
55
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