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review derivation: velocity at distance r of object dropped from infinity

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Notes for this derivation:
https://www.youtube.com/watch?v=5F1XcTjpJs4 - Derivation of Gravitational Potential Energy by Rhett Allain

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
5 substitute LHS of expr 1 into expr 2
  1. 5902985919; locally 3470082:
    F=Gm1m2x2x^
  2. 7882872592; locally 6798426:
    Wto system=rFdr
  1. 3566149658; locally 7300369:
    Wto system=rGm1m2x2dx
failed 5902985919:
7882872592:
3566149658:
5902985919:
7882872592:
3566149658:
2 declare initial expr
  1. 5902985919; locally 3470082:
    F=Gm1m2x2x^
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
  1. 8049905441; locally 9781919:
    ΔKE=KEfinalKEinitial
  2. 1114820451; locally 9835406:
    Wby system=ΔKE
  1. 5779256336; locally 8118190:
    Wby system=KEfinalKEinitial
valid 8049905441:
1114820451:
5779256336:
8049905441:
1114820451:
5779256336:
15 declare initial expr
  1. 2924222857; locally 1712972:
    vinitial=v(r=)
no validation is available for declarations 2924222857:
2924222857:
8 simplify
  1. 5596822289; locally 5818573:
    Wto system=Gm1m2(1x|r)
  1. 2061086175; locally 2429271:
    Wto system=Gm1m2(1r1)
LHS diff is 0 RHS diff is pdg5022*pdg6277*(-pdg4851 + pdg4851(-1/pdg2530)) 5596822289:
2061086175:
5596822289:
2061086175:
11 declare initial expr
  1. 8357234146; locally 5104592:
    KE=12mv2
no validation is available for declarations 8357234146:
8357234146:
26 declare final expr
  1. 2005061870; locally 3435796:
    v(r)=2Gm2r
no validation is available for declarations 2005061870:
2005061870:
7 evaluate definite integral
  1. 8405272745; locally 9707318:
    Wto system=Gm1m2r1x2dx
  1. 5596822289; locally 5818573:
    Wto system=Gm1m2(1x|r)
LHS diff is 0 RHS diff is pdg4851*pdg5022*pdg6277*(1 + 1/pdg2530) 8405272745:
5596822289:
8405272745:
5596822289:
9 simplify
  1. 2061086175; locally 2429271:
    Wto system=Gm1m2(1r1)
  1. 4393670960; locally 4947999:
    Wto system=Gm1m2r
LHS diff is 0 RHS diff is pdg5022*pdg6277*(-pdg2530*pdg4851(-1/pdg2530) - pdg4851)/pdg2530 2061086175:
4393670960:
2061086175:
4393670960:
25 change variable X to Y
  1. 5846639423; locally 7112224:
    vfinal=2Gm2r
  1. 6599829782:
    vfinal
  2. 3531380618:
    v(r)
  1. 2005061870; locally 3435796:
    v(r)=2Gm2r
valid 5846639423:
2005061870:
5846639423:
2005061870:
6 simplify
  1. 3566149658; locally 7300369:
    Wto system=rGm1m2x2dx
  1. 8405272745; locally 9707318:
    Wto system=Gm1m2r1x2dx
valid 3566149658:
8405272745:
3566149658:
8405272745:
16 substitute LHS of expr 1 into expr 2
  1. 3214170322; locally 8462685:
    v(r=)=0
  2. 2924222857; locally 1712972:
    vinitial=v(r=)
  1. 2998709778; locally 6923850:
    vinitial=0
Nothing to split 3214170322: no LHS/RHS split
2924222857:
2998709778:
3214170322: N/A
2924222857:
2998709778:
3 declare initial expr
  1. 1114820451; locally 9835406:
    Wby system=ΔKE
no validation is available for declarations 1114820451:
1114820451:
18 substitute LHS of expr 1 into expr 2
  1. 9510328252; locally 7110498:
    KEinitial=0
  2. 5779256336; locally 8118190:
    Wby system=KEfinalKEinitial
  1. 5850144586; locally 2751634:
    Wby system=KEfinal
valid 9510328252:
5779256336:
5850144586:
9510328252:
5779256336:
5850144586:
10 declare initial expr
  1. 8049905441; locally 9781919:
    ΔKE=KEfinalKEinitial
no validation is available for declarations 8049905441:
8049905441:
13 change three variables in expr
  1. 8357234146; locally 5104592:
    KE=12mv2
  1. 3731774096:
    KE
  2. 3350802342:
    KEinitial
  3. 5904227750:
    m
  4. 6281834543:
    m1
  5. 8066819515:
    v
  6. 3274176452:
    vinitial
  1. 6091977310; locally 9031887:
    KEinitial=12m1vinitial2
valid 8357234146:
6091977310:
8357234146:
6091977310:
21 substitute LHS of expr 1 into expr 2
  1. 9081138616; locally 6536576:
    Wby system=12m1vfinal2
  2. 2907404069; locally 2619766:
    Wby system=Wto system
  1. 4947831649; locally 8655239:
    12m1vfinal2=Wto system
valid 9081138616:
2907404069:
4947831649:
9081138616:
2907404069:
4947831649:
20 declare initial expr
  1. 2907404069; locally 2619766:
    Wby system=Wto system
no validation is available for declarations 2907404069:
2907404069:
22 substitute LHS of expr 1 into expr 2
  1. 4393670960; locally 4947999:
    Wto system=Gm1m2r
  2. 4947831649; locally 8655239:
    12m1vfinal2=Wto system
  1. 6892595652; locally 2942416:
    12m1vfinal2=Gm1m2r
valid 4393670960:
4947831649:
6892595652:
4393670960:
4947831649:
6892595652:
17 substitute LHS of expr 1 into expr 2
  1. 2998709778; locally 6923850:
    vinitial=0
  2. 6091977310; locally 9031887:
    KEinitial=12m1vinitial2
  1. 9510328252; locally 7110498:
    KEinitial=0
valid 2998709778:
6091977310:
9510328252:
2998709778:
6091977310:
9510328252:
14 change three variables in expr
  1. 8357234146; locally 5104592:
    KE=12mv2
  1. 4587046017:
    KE
  2. 3939572542:
    KEfinal
  3. 9350720370:
    m
  4. 3166466250:
    m1
  5. 6038673136:
    v
  6. 1616666229:
    vfinal
  1. 8552710882; locally 1397156:
    KEfinal=12m1vfinal2
failed 8357234146:
8552710882:
8357234146:
8552710882:
4 declare initial expr
  1. 7882872592; locally 6798426:
    Wto system=rFdr
no validation is available for declarations 7882872592:
7882872592:
24 square root both sides
  1. 7112646057; locally 4594601:
    vfinal2=2Gm2r
  1. 5846639423; locally 7112224:
    vfinal=2Gm2r
  2. 5693047217; locally 1366396:
    vfinal=2Gm2r
no check performed 7112646057:
5846639423:
5693047217:
7112646057:
5846639423:
5693047217:
19 substitute LHS of expr 1 into expr 2
  1. 8552710882; locally 1397156:
    KEfinal=12m1vfinal2
  2. 5850144586; locally 2751634:
    Wby system=KEfinal
  1. 9081138616; locally 6536576:
    Wby system=12m1vfinal2
valid 8552710882:
5850144586:
9081138616:
8552710882:
5850144586:
9081138616:
23 multiply both sides by
  1. 6892595652; locally 2942416:
    12m1vfinal2=Gm1m2r
  1. 7410526982:
    2/m1
  1. 7112646057; locally 4594601:
    vfinal2=2Gm2r
valid 6892595652:
7112646057:
6892595652:
7112646057:
1 declare initial expr
  1. 3214170322; locally 8462685:
    v(r=)=0
no validation is available for declarations 3214170322: no LHS/RHS split
3214170322: N/A
starting velocity at infinity is zero
Physics Derivation Graph: Steps for velocity at distance r of object dropped from infinity

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
4851 variable m_2
m2
real
  • mass: 1
mass 31
5734 variable \Delta KE
ΔKE
real dimensionless change in kinetic energy 2
6191 variable W_{\rm by\ system}
Wby system
real
  • length: 2
  • mass: 1
  • time: -2
work done by system 5
4929 variable KE
KE
['real']
  • length: 2
  • mass: 1
  • time: -2
kinetic energy 7
9372 variable W_{\rm to\ system}
Wto system
real
  • length: 2
  • mass: 1
  • time: -2
work done to system 8
1934 variable v_{\rm initial}
vinitial
real
  • length: 1
  • time: -1
initial velocity 4
8909 variable v_{\rm final}
vfinal
real
  • length: 1
  • time: -1
final velocity 9
1357 variable v
v
['real']
  • length: 1
  • time: -1
velocity 83
5156 variable m
m
['real']
  • mass: 1
mass 69
5022 variable m_1
m1
real
  • mass: 1
mass 35
6277 constant G
G
real
  • length: 3
  • mass: -1
  • time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
60
4202 variable F
F
['real']
  • length: 1
  • mass: 1
  • time: -2
force 21
5340 variable KE_{\rm final}
KEfinal
real dimensionless final kinetic energy 5
4121 variable KE_{\rm initial}
KEinitial
real dimensionless initial kinetic energy 5
2530 variable r
r
['real']
  • length: 1
radius 60
4037 variable x
x
['real']
  • length: 1
position 53
6777 variable \vec{F}
F
real dimensionless force 1
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