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review derivation: escape velocity

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
20 swap LHS with RHS
  1. 2977457786; locally 3358651:
    \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2\)
    \(\frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}} = pdg_{8656}^{2}\)
  1. 9412953728; locally 3908344:
    \(v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)
    \(pdg_{8656}^{2} = \frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
valid 2977457786:
9412953728:
2977457786:
9412953728:
8 substitute LHS of two expressions into expr
  1. 4303372136; locally 6310702:
    \(E_1 = KE_1 + PE_1\)
    \(pdg_{5579} = pdg_{1955} + pdg_{4093}\)
  2. 7875206161; locally 5160388:
    \(E_2 = KE_2 + PE_2\)
    \(pdg_{4550} = pdg_{1352} + pdg_{8849}\)
  3. 8558338742; locally 6330719:
    \(E_2 = E_1\)
    \(pdg_{4550} = pdg_{5579}\)
  1. 8960645192; locally 4840471:
    \(KE_2 + PE_2 = KE_1 + PE_1\)
    \(pdg_{1552} + pdg_{8849} = pdg_{1955} + pdg_{4093}\)
failed 4303372136:
7875206161:
8558338742:
8960645192:
4303372136:
7875206161:
8558338742:
8960645192:
9 declare assumption
  1. 2267521164; locally 7682341:
    \(PE_2 = 0\)
    \(pdg_{8849} = 0\)
no validation is available for declarations 2267521164:
2267521164:
13 declare initial expr
  1. 7573835180; locally 6773616:
    \(PE_{\rm Earth\ surface} = -W\)
    \(pdg_{6431} = - pdg_{6789}\)
no validation is available for declarations 7573835180:
7573835180:
19 multiply both sides by
  1. 1143343287; locally 7567097:
    \(G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2\)
    \(\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{8656}^{2}}{2}\)
  1. 5775658332:
    \(2\)
    \(2\)
  1. 2977457786; locally 3358651:
    \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2\)
    \(\frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}} = pdg_{8656}^{2}\)
valid 1143343287:
2977457786:
1143343287:
2977457786:
4 evaluate definite integral
  1. 4447113478; locally 4803506:
    \(\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx\)
    \(\int 1\, dpdg_{6789} = pdg_{4851} pdg_{5022} pdg_{6277} \int\limits_{pdg_{3236}}^{infty} \frac{1}{pdg_{4037}^{2}}\, dpdg_{4037}\)
  1. 5732331610; locally 1089445:
    \(W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
    \(pdg_{6277}\)
Nothing to split 4447113478:
5732331610:
4447113478:
5732331610:
7 simplify
  1. 5978756813; locally 2190752:
    \(W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)
    \(pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
  1. 7749253510; locally 2238158:
    \(W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)
    \(pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
valid 5978756813:
7749253510:
5978756813:
7749253510:
22 declare final expr
  1. 1330874553; locally 6389964:
    \(v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
    \(pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}\)
no validation is available for declarations 1330874553:
1330874553:
23 change two variables in expr
  1. 1330874553; locally 6389964:
    \(v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
    \(pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}\)
  1. 2674546234:
    \(m_{\rm Earth}\)
    \(pdg_{5458}\)
  2. 2135482543:
    \(m\)
    \(pdg_{5156}\)
  3. 2396787389:
    \(r_{\rm Earth}\)
    \(pdg_{3236}\)
  4. 8020058613:
    \(r\)
    \(pdg_{2530}\)
  1. 5404822208; locally 1619188:
    \(v_{\rm escape} = \sqrt{2 G \frac{m}{r}}\)
    \(pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5156} pdg_{6277}}{pdg_{2530}}}\)
valid 1330874553:
5404822208:
1330874553:
5404822208:
replaced Earth-specific variables
21 square root both sides
  1. 9412953728; locally 3908344:
    \(v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)
    \(pdg_{8656}^{2} = \frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
  1. 1330874553; locally 6389964:
    \(v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
    \(pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}\)
  2. 2750380042; locally 8779043:
    \(v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
    \(pdg_{8656} = - \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}\)
no check performed 9412953728:
1330874553:
2750380042:
9412953728:
1330874553:
2750380042:
2 substitute LHS of expr 1 into expr 2
  1. 6935745841; locally 3279838:
    \(F = G \frac{m_1 m_2}{x^2}\)
    \(pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}\)
  2. 1590774089; locally 2123766:
    \(dW = F dx\)
    \(pdg_{9398} = pdg_{4202} pdg_{9199}\)
  1. 8604483515; locally 3686928:
    \(dW = G \frac{m_1 m_2}{x^2} dx\)
    \(pdg_{9398} = \frac{pdg_{4851} pdg_{5022} pdg_{6277} pdg_{9199}}{pdg_{4037}^{2}}\)
valid 6935745841:
1590774089:
8604483515:
6935745841:
1590774089:
8604483515:
18 simplify
  1. 9703482302; locally 6523887:
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2\)
    \(\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}\)
  1. 1143343287; locally 7567097:
    \(G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2\)
    \(\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{8656}^{2}}{2}\)
LHS diff is pdg5458*pdg6277*(pdg5156 - 1)/pdg3236 RHS diff is pdg8656**2*(pdg5156 - 1)/2 9703482302:
1143343287:
9703482302:
1143343287:
3 integrate
  1. 8604483515; locally 3686928:
    \(dW = G \frac{m_1 m_2}{x^2} dx\)
    \(pdg_{9398} = \frac{pdg_{4851} pdg_{5022} pdg_{6277} pdg_{9199}}{pdg_{4037}^{2}}\)
  1. 4447113478; locally 4803506:
    \(\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx\)
    \(\int 1\, dpdg_{6789} = pdg_{4851} pdg_{5022} pdg_{6277} \int\limits_{pdg_{3236}}^{infty} \frac{1}{pdg_{4037}^{2}}\, dpdg_{4037}\)
no check performed 8604483515:
4447113478:
8604483515:
4447113478:
14 substitute LHS of expr 1 into expr 2
  1. 7749253510; locally 2238158:
    \(W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)
    \(pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
  2. 7573835180; locally 6773616:
    \(PE_{\rm Earth\ surface} = -W\)
    \(pdg_{6431} = - pdg_{6789}\)
  1. 3846041519; locally 9437784:
    \(PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
    \(pdg_{6431} = - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
valid 7749253510:
7573835180:
3846041519:
7749253510:
7573835180:
3846041519:
1 declare initial expr
  1. 6935745841; locally 3279838:
    \(F = G \frac{m_1 m_2}{x^2}\)
    \(pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}\)
no validation is available for declarations 6935745841:
6935745841:
10 declare assumption
  1. 1840080113; locally 9324316:
    \(KE_2 = 0\)
    \(pdg_{1552} = 0\)
no validation is available for declarations 1840080113:
1840080113:
11 substitute LHS of two expressions into expr
  1. 2267521164; locally 7682341:
    \(PE_2 = 0\)
    \(pdg_{8849} = 0\)
  2. 1840080113; locally 9324316:
    \(KE_2 = 0\)
    \(pdg_{1552} = 0\)
  3. 8960645192; locally 4840471:
    \(KE_2 + PE_2 = KE_1 + PE_1\)
    \(pdg_{1552} + pdg_{8849} = pdg_{1955} + pdg_{4093}\)
  1. 9749777192; locally 8369684:
    \(0 = KE_1 + PE_1\)
    \(0 = pdg_{1955} + pdg_{4093}\)
failed 2267521164:
1840080113:
8960645192:
9749777192: inconsistent dimensions
 2267521164:
1840080113:
8960645192:
9749777192: N/A
5 change two variables in expr
  1. 5732331610; locally 1089445:
    \(W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
    \(pdg_{6277}\)
  1. 1413137236:
    \(m_1\)
    \(pdg_{5022}\)
  2. 9072369552:
    \(m_{\rm Earth}\)
    \(pdg_{5458}\)
  3. 2764966428:
    \(m_2\)
    \(pdg_{4851}\)
  4. 7140470627:
    \(m\)
    \(pdg_{5156}\)
  1. 6131764194; locally 2341415:
    \(W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
    \(W = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{4037}^{2}}\)
Nothing to split 5732331610:
6131764194:
5732331610:
6131764194:
16 substitute LHS of two expressions into expr
  1. 6870322215; locally 5106827:
    \(KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2\)
    \(pdg_{5332} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}\)
  2. 3846041519; locally 9437784:
    \(PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
    \(pdg_{6431} = - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
  3. 2503972039; locally 9967559:
    \(0 = KE_{\rm escape} + PE_{\rm Earth\ surface}\)
    \(0 = pdg_{5332} + pdg_{6431}\)
  1. 2042298788; locally 3493665:
    \(0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2\)
    \(0 = \frac{pdg_{5156} pdg_{8656}^{2}}{2} - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
failed 6870322215:
3846041519:
2503972039: inconsistent dimensions
2042298788:
6870322215:
3846041519:
2503972039: N/A
2042298788:
15 change two variables in expr
  1. 8357234146; locally 3778087:
    \(KE = \frac{1}{2} m v^2\)
    \(pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}\)
  1. 5021965469:
    \(KE\)
    \(pdg_{4929}\)
  2. 9370882921:
    \(KE_{\rm escape}\)
    \(pdg_{5332}\)
  3. 6681646197:
    \(v\)
    \(pdg_{1357}\)
  4. 6498985149:
    \(v_{\rm escape}\)
    \(pdg_{8656}\)
  1. 6870322215; locally 5106827:
    \(KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2\)
    \(pdg_{5332} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}\)
valid 8357234146:
6870322215:
8357234146:
6870322215:
17 add X to both sides
  1. 2042298788; locally 3493665:
    \(0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2\)
    \(0 = \frac{pdg_{5156} pdg_{8656}^{2}}{2} - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
  1. 5050429607:
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
    \(\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
  1. 9703482302; locally 6523887:
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2\)
    \(\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}\)
valid 2042298788:
9703482302:
2042298788:
9703482302:
6 simplify
  1. 6131764194; locally 2341415:
    \(W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
    \(W = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{4037}^{2}}\)
  1. 5978756813; locally 2190752:
    \(W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)
    \(pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}\)
LHS diff is W - pdg6789 RHS diff is pdg5156*pdg5458*pdg6277*(pdg3236 - pdg4037**2)/(pdg3236*pdg4037**2) 6131764194:
5978756813:
6131764194:
5978756813:
12 change two variables in expr
  1. 9749777192; locally 8369684:
    \(0 = KE_1 + PE_1\)
    \(0 = pdg_{1955} + pdg_{4093}\)
  1. 5591692598:
    \(KE_1\)
    \(pdg_{1955}\)
  2. 8416464049:
    \(KE_{\rm escape}\)
    \(pdg_{5332}\)
  3. 6158970683:
    \(PE_1\)
    \(pdg_{4093}\)
  4. 8871333437:
    \(PE_{\rm Earth\ surface}\)
    \(pdg_{6431}\)
  1. 2503972039; locally 9967559:
    \(0 = KE_{\rm escape} + PE_{\rm Earth\ surface}\)
    \(0 = pdg_{5332} + pdg_{6431}\)
valid 9749777192: inconsistent dimensions
2503972039: inconsistent dimensions
 9749777192: N/A
2503972039: N/A
Physics Derivation Graph: Steps for escape velocity

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
9398 variable dW
\(dW\)		 real
  • length: 2
  • mass: 1
  • time: -2
 differential work
  • str_note
2
4550 variable E_2
\(E_2\)		 real
  • length: 2
  • mass: 1
  • time: -2
 energy 2
  • str_note
9
9199 variable dx
\(dx\)		 ['real']
  • length: 1
 15
3236 constant r_{\rm Earth}
\(r_{\rm Earth}\)		 real
  • length: 1
 radius of Earth 6.3781*10^{6}   m
 21
4093 variable PE_1
\(PE_1\)		 real
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 12
1352 variable KE_2
\(KE_2\)		 real
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 14
4851 variable m_2
\(m_2\)		 real
  • mass: 1
 mass 31
6431 variable PE_{\rm Earth\ surface}
\(PE_{\rm Earth\ surface}\)		 real
  • length: 2
  • mass: 1
  • time: -2
 potential energy at the Earth's surface
  • str_note
4
1955 variable KE_1
\(KE_1\)		 real
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 17
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
5458 constant m_{\rm Earth}
\(m_{\rm Earth}\)		 real
  • mass: 2
 mass of Earth 5.97237*10^24   kg
 34
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
5332 variable KE_{\rm escape}
\(KE_{\rm escape}\)		 real
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy of escape velocity
  • str_note
4
5579 variable E_1
\(E_1\)		 real
  • length: 2
  • mass: 1
  • time: -2
 energy 1
  • str_note
9
4202 variable F
\(F\)		 ['real']
  • length: 1
  • mass: 1
  • time: -2
 force 21
6789 variable W
\(W\)		 real
  • length: 2
  • mass: 1
  • time: -2
 work 10
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 60
8849 variable PE_2
\(PE_2\)		 real
  • length: 2
  • mass: 1
  • time: -2
 potential energy 11
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
1552 variable j
\(j\)		 ['integer'] dimensionless index 7
5022 variable m_1
\(m_1\)		 real
  • mass: 1
 mass 35
2530 variable r
\(r\)		 ['real']
  • length: 1
 radius 60
4929 variable KE
\(KE\)		 ['real']
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 7
8656 variable v_{\rm escape}
\(v_{\rm escape}\)		 real
  • length: 1
  • time: -1
 escape velocity 12
MESSAGES: