This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.
Notes for this derivation:
https://www.youtube.com/watch?v=5F1XcTjpJs4 - Derivation of Gravitational Potential Energy by Rhett Allain
Index | Inference Rule | Input latex | Feeds latex | Output latex | step validity | dimension check | unit check | notes |
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5 | substitute LHS of expr 1 into expr 2 |
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failed |
5902985919:
7882872592: 3566149658: |
5902985919:
7882872592: 3566149658: |
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2 | declare initial expr |
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no validation is available for declarations |
5902985919:
|
5902985919:
|
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form |
12 | substitute LHS of expr 1 into expr 2 |
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valid |
8049905441:
1114820451: 5779256336: |
8049905441:
1114820451: 5779256336: |
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15 | declare initial expr |
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no validation is available for declarations |
2924222857:
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2924222857:
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8 | simplify |
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LHS diff is 0 RHS diff is pdg5022*pdg6277*(-pdg4851 + pdg4851(-1/pdg2530)) |
5596822289:
2061086175: |
5596822289:
2061086175: |
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11 | declare initial expr |
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no validation is available for declarations |
8357234146:
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8357234146:
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26 | declare final expr |
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no validation is available for declarations |
2005061870:
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2005061870:
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7 | evaluate definite integral |
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LHS diff is 0 RHS diff is pdg4851*pdg5022*pdg6277*(1 + 1/pdg2530) |
8405272745:
5596822289: |
8405272745:
5596822289: |
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9 | simplify |
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LHS diff is 0 RHS diff is -pdg5022*pdg6277*(pdg2530*pdg4851(-1/pdg2530) + pdg4851)/pdg2530 |
2061086175:
4393670960: |
2061086175:
4393670960: |
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25 | change variable X to Y |
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valid |
5846639423:
2005061870: |
5846639423:
2005061870: |
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6 | simplify |
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valid |
3566149658:
8405272745: |
3566149658:
8405272745: |
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16 | substitute LHS of expr 1 into expr 2 |
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Nothing to split |
3214170322:
no LHS/RHS split 2924222857: 2998709778: |
3214170322:
N/A 2924222857: 2998709778: |
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3 | declare initial expr |
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no validation is available for declarations |
1114820451:
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1114820451:
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18 | substitute LHS of expr 1 into expr 2 |
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valid |
9510328252:
5779256336: 5850144586: |
9510328252:
5779256336: 5850144586: |
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10 | declare initial expr |
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no validation is available for declarations |
8049905441:
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8049905441:
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13 | change three variables in expr |
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valid |
8357234146:
6091977310: |
8357234146:
6091977310: |
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21 | substitute LHS of expr 1 into expr 2 |
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valid |
9081138616:
2907404069: 4947831649: |
9081138616:
2907404069: 4947831649: |
|
20 | declare initial expr |
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no validation is available for declarations |
2907404069:
|
2907404069:
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22 | substitute LHS of expr 1 into expr 2 |
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valid |
4393670960:
4947831649: 6892595652: |
4393670960:
4947831649: 6892595652: |
|
17 | substitute LHS of expr 1 into expr 2 |
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valid |
2998709778:
6091977310: 9510328252: |
2998709778:
6091977310: 9510328252: |
|
14 | change three variables in expr |
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failed |
8357234146:
8552710882: |
8357234146:
8552710882: |
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4 | declare initial expr |
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no validation is available for declarations |
7882872592:
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7882872592:
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24 | square root both sides |
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no check performed |
7112646057:
5846639423: 5693047217: |
7112646057:
5846639423: 5693047217: |
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19 | substitute LHS of expr 1 into expr 2 |
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valid |
8552710882:
5850144586: 9081138616: |
8552710882:
5850144586: 9081138616: |
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23 | multiply both sides by |
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valid |
6892595652:
7112646057: |
6892595652:
7112646057: |
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1 | declare initial expr |
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|
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no validation is available for declarations |
3214170322:
no LHS/RHS split |
3214170322:
N/A |
starting velocity at infinity is zero |
symbol ID | category | latex | scope | dimension | name | value | Used in derivations | references | |
---|---|---|---|---|---|---|---|---|---|
4202 | variable |
F
\(F\) |
['real'] |
|
force | 21 | |||
8909 | variable |
v_{\rm final}
\(v_{\rm final}\) |
real |
|
final velocity | 9 | |||
2530 | variable |
r
\(r\) |
['real'] |
|
radius | 34 | |||
9372 | variable |
W_{\rm to\ system}
\(W_{\rm to\ system}\) |
real |
|
work done to system | 8 | |||
5340 | variable |
KE_{\rm final}
\(KE_{\rm final}\) |
real | dimensionless | final kinetic energy | 5 | |||
4037 | variable |
x
\(x\) |
['real'] |
|
position |
|
53 | ||
1357 | variable |
v
\(v\) |
['real'] |
|
velocity |
|
80 | ||
6277 | constant |
G
\(G\) |
real |
|
gravitational constant |
6.67430*10^{-11} m^3 * kg^-1 * s^-2 |
46 | ||
1934 | variable |
v_{\rm initial}
\(v_{\rm initial}\) |
real |
|
initial velocity | 4 | |||
5734 | variable |
\Delta KE
\(\Delta KE\) |
real | dimensionless | change in kinetic energy | 2 | |||
4929 | variable |
KE
\(KE\) |
['real'] |
|
kinetic energy | 7 | |||
4851 | variable |
m_2
\(m_2\) |
real |
|
mass | 17 | |||
6191 | variable |
W_{\rm by\ system}
\(W_{\rm by\ system}\) |
real |
|
work done by system | 5 | |||
5022 | variable |
m_1
\(m_1\) |
real |
|
mass | 18 | |||
4121 | variable |
KE_{\rm initial}
\(KE_{\rm initial}\) |
real | dimensionless | initial kinetic energy | 5 | |||
5156 | variable |
m
\(m\) |
['real'] |
|
mass |
|
57 | ||
6777 | variable |
\vec{F}
\(\vec{F}\) |
real | dimensionless | force | 1 |