This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.
Notes for this derivation:
Index | Inference Rule | Input latex | Feeds latex | Output latex | step validity | dimension check | unit check | notes |
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20 | swap LHS with RHS |
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valid |
2977457786:
9412953728: |
2977457786:
9412953728: |
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8 | substitute LHS of two expressions into expr |
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failed |
4303372136:
7875206161: 8558338742: 8960645192: |
4303372136:
7875206161: 8558338742: 8960645192: |
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9 | declare assumption |
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no validation is available for declarations |
2267521164:
|
2267521164:
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13 | declare initial expr |
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no validation is available for declarations |
7573835180:
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7573835180:
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19 | multiply both sides by |
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valid |
1143343287:
2977457786: |
1143343287:
2977457786: |
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4 | evaluate definite integral |
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Nothing to split |
4447113478:
5732331610: |
4447113478:
5732331610: |
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7 | simplify |
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valid |
5978756813:
7749253510: |
5978756813:
7749253510: |
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22 | declare final expr |
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no validation is available for declarations |
1330874553:
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1330874553:
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23 | change two variables in expr |
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valid |
1330874553:
5404822208: |
1330874553:
5404822208: |
replaced Earth-specific variables |
21 | square root both sides |
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no check performed |
9412953728:
1330874553: 2750380042: |
9412953728:
1330874553: 2750380042: |
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2 | substitute LHS of expr 1 into expr 2 |
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valid |
6935745841:
1590774089: 8604483515: |
6935745841:
1590774089: 8604483515: |
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18 | simplify |
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LHS diff is pdg5458*pdg6277*(pdg5156 - 1)/pdg3236 RHS diff is pdg8656**2*(pdg5156 - 1)/2 |
9703482302:
1143343287: |
9703482302:
1143343287: |
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3 | integrate |
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no check performed |
8604483515:
4447113478: |
8604483515:
4447113478: |
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14 | substitute LHS of expr 1 into expr 2 |
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valid |
7749253510:
7573835180: 3846041519: |
7749253510:
7573835180: 3846041519: |
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1 | declare initial expr |
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no validation is available for declarations |
6935745841:
|
6935745841:
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10 | declare assumption |
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no validation is available for declarations |
1840080113:
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1840080113:
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11 | substitute LHS of two expressions into expr |
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failed |
2267521164:
1840080113: 8960645192: 9749777192: inconsistent dimensions |
2267521164:
1840080113: 8960645192: 9749777192: N/A |
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5 | change two variables in expr |
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Nothing to split |
5732331610:
6131764194: |
5732331610:
6131764194: |
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16 | substitute LHS of two expressions into expr |
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failed |
6870322215:
3846041519: 2503972039: inconsistent dimensions 2042298788: |
6870322215:
3846041519: 2503972039: N/A 2042298788: |
|
15 | change two variables in expr |
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valid |
8357234146:
6870322215: |
8357234146:
6870322215: |
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17 | add X to both sides |
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valid |
2042298788:
9703482302: |
2042298788:
9703482302: |
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6 | simplify |
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LHS diff is W - pdg6789 RHS diff is pdg5156*pdg5458*pdg6277*(pdg3236 - pdg4037**2)/(pdg3236*pdg4037**2) |
6131764194:
5978756813: |
6131764194:
5978756813: |
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12 | change two variables in expr |
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valid |
9749777192:
inconsistent dimensions 2503972039: inconsistent dimensions |
9749777192:
N/A 2503972039: N/A |
symbol ID | category | latex | scope | dimension | name | value | Used in derivations | references | |
---|---|---|---|---|---|---|---|---|---|
4851 | variable |
m_2
\(m_2\) |
real |
|
mass | 31 | |||
9398 | variable |
dW
\(dW\) |
real |
|
differential work |
|
2 | ||
4550 | variable |
E_2
\(E_2\) |
real |
|
energy 2 |
|
9 | ||
1955 | variable |
KE_1
\(KE_1\) |
real |
|
kinetic energy | 17 | |||
6277 | constant |
G
\(G\) |
real |
|
gravitational constant |
6.67430*10^{-11} m^3 * kg^-1 * s^-2 |
|
60 | |
8656 | variable |
v_{\rm escape}
\(v_{\rm escape}\) |
real |
|
escape velocity | 12 | |||
3236 | constant |
r_{\rm Earth}
\(r_{\rm Earth}\) |
real |
|
radius of Earth |
6.3781*10^{6} m |
21 | ||
1352 | variable |
KE_2
\(KE_2\) |
real |
|
kinetic energy | 14 | |||
5022 | variable |
m_1
\(m_1\) |
real |
|
mass | 35 | |||
4929 | variable |
KE
\(KE\) |
['real'] |
|
kinetic energy | 7 | |||
1357 | variable |
v
\(v\) |
['real'] |
|
velocity |
|
83 | ||
9199 | variable |
dx
\(dx\) |
['real'] |
|
15 | ||||
5156 | variable |
m
\(m\) |
['real'] |
|
mass |
|
69 | ||
6431 | variable |
PE_{\rm Earth\ surface}
\(PE_{\rm Earth\ surface}\) |
real |
|
potential energy at the Earth's surface |
|
4 | ||
5458 | constant |
m_{\rm Earth}
\(m_{\rm Earth}\) |
real |
|
mass of Earth |
5.97237*10^24 kg |
34 | ||
5332 | variable |
KE_{\rm escape}
\(KE_{\rm escape}\) |
real |
|
kinetic energy of escape velocity |
|
4 | ||
4037 | variable |
x
\(x\) |
['real'] |
|
position |
|
53 | ||
2530 | variable |
r
\(r\) |
['real'] |
|
radius | 60 | |||
6789 | variable |
W
\(W\) |
real |
|
work | 10 | |||
1552 | variable |
j
\(j\) |
['integer'] | dimensionless | index | 7 | |||
8849 | variable |
PE_2
\(PE_2\) |
real |
|
potential energy | 11 | |||
4202 | variable |
F
\(F\) |
['real'] |
|
force | 21 | |||
5579 | variable |
E_1
\(E_1\) |
real |
|
energy 1 |
|
9 | ||
4093 | variable |
PE_1
\(PE_1\) |
real |
|
kinetic energy | 12 |