## 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: error for dim with 3472836147
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: error for dim with 3472836147
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

symbol ID category latex scope dimension name value Used in derivations references
2530 variable r
$$r$$
['real']
• length: 1
60
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
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
3141 constant \pi
$$\pi$$
['real'] dimensionless pi 3.1415   dimensionless
72
6081 constant r_{\rm Earth\ orbit}
$$r_{\rm Earth\ orbit}$$
['real']
• length: 1
radius of Earth's orbit 1.496*10^8   km
5
5344 variable t_{\rm Earth\ orbit}
$$t_{\rm Earth\ orbit}$$
real
• time: 1
time of Earth's orbit around sun
• str_note
5
MESSAGE:
• local variable 'all_df' referenced before assignment