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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
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:
4180845508:
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:
12 simplify
  1. 6998364753; locally 8698819:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 pdg_{3141}\)
  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*pdg3141 - 29.8 6998364753: inconsistent dimensions
4180845508:
6998364753: N/A
4180845508:
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:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 pdg_{3141} pdg_{6081}\)
LHS diff is 0 RHS diff is 0.0337552742616034*pdg3141*pdg6081 8721295221:
3080027960:
4593428198: inconsistent dimensions
 8721295221:
3080027960:
4593428198: N/A
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
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: inconsistent dimensions
 3472836147: N/A
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:
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:
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:
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:
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:
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:
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:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 pdg_{3141} pdg_{6081}\)
  1. 6998364753; locally 8698819:
    \(v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}\)
    \(pdg_{7427} = 0.632911392405063 pdg_{3141}\)
LHS diff is 0 RHS diff is 0.313924050632911*pdg3141 3472836147: inconsistent dimensions
4593428198: inconsistent dimensions
6998364753: inconsistent dimensions
 3472836147: N/A
4593428198: N/A
6998364753: N/A
Physics Derivation Graph: Steps for speed of Earth around Sun

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
3141 constant \pi
\(\pi\)		 ['real'] dimensionless pi 3.1415   dimensionless
 72
5344 variable t_{\rm Earth\ orbit}
\(t_{\rm Earth\ orbit}\)		 real
  • time: 1
 time of Earth's orbit around sun
  • str_note
5
1534 variable C_{\rm Earth\ orbit}
\(C_{\rm Earth\ orbit}\)		 real
  • length: 1
 circumference of Earth's orbit 4
3034 variable C
\(C\)		 ['real']
  • length: 1
 circumference 5
1943 variable d
\(d\)		 ['real']
  • length: 1
 displacement 25
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
1467 variable t
\(t\)		 ['real']
  • time: 1
 time 121
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
2530 variable r
\(r\)		 ['real']
  • length: 1
 radius 60
6081 constant r_{\rm Earth\ orbit}
\(r_{\rm Earth\ orbit}\)		 ['real']
  • length: 1
 radius of Earth's orbit 1.496*10^8   km
 5
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