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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:
    \(\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}\)
    \(\)
  2. 7882872592; locally 6798426:
    \(W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}\)
    \(\)
  1. 3566149658; locally 7300369:
    \(W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
    \(\)
failed 5902985919:
7882872592:
3566149658:
5902985919:
7882872592:
3566149658:
2 declare initial expr
  1. 5902985919; locally 3470082:
    \(\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}\)
    \(\)
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:
    \(\Delta KE = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
  2. 1114820451; locally 9835406:
    \(W_{\rm by\ system} = \Delta KE\)
    \(\)
  1. 5779256336; locally 8118190:
    \(W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
valid 8049905441:
1114820451:
5779256336:
8049905441:
1114820451:
5779256336:
15 declare initial expr
  1. 2924222857; locally 1712972:
    \(v_{\rm initial} = v(r=\infty)\)
    \(\)
no validation is available for declarations 2924222857:
2924222857:
8 simplify
  1. 5596822289; locally 5818573:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
    \(\)
  1. 2061086175; locally 2429271:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
    \(\)
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:
    \(KE = \frac{1}{2} m v^2\)
    \(\)
no validation is available for declarations 8357234146:
8357234146:
26 declare final expr
  1. 2005061870; locally 3435796:
    \(v(r) = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
no validation is available for declarations 2005061870:
2005061870:
7 evaluate definite integral
  1. 8405272745; locally 9707318:
    \(W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
    \(\)
  1. 5596822289; locally 5818573:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
    \(\)
LHS diff is 0 RHS diff is pdg4851*pdg5022*pdg6277*(1 + 1/pdg2530) 8405272745:
5596822289:
8405272745:
5596822289:
9 simplify
  1. 2061086175; locally 2429271:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
    \(\)
  1. 4393670960; locally 4947999:
    \(W_{\rm to\ system} = \frac{G m_1 m_2}{r}\)
    \(\)
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:
    \(v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
  1. 6599829782:
    \(v_{\rm final}\)
    \(\)
  2. 3531380618:
    \(v(r)\)
    \(\)
  1. 2005061870; locally 3435796:
    \(v(r) = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
valid 5846639423:
2005061870:
5846639423:
2005061870:
6 simplify
  1. 3566149658; locally 7300369:
    \(W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
    \(\)
  1. 8405272745; locally 9707318:
    \(W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
    \(\)
valid 3566149658:
8405272745:
3566149658:
8405272745:
16 substitute LHS of expr 1 into expr 2
  1. 3214170322; locally 8462685:
    \(v(r=\infty) = 0\)
    \(\)
  2. 2924222857; locally 1712972:
    \(v_{\rm initial} = v(r=\infty)\)
    \(\)
  1. 2998709778; locally 6923850:
    \(v_{\rm initial} = 0\)
    \(\)
Nothing to split 3214170322: no LHS/RHS split
2924222857:
2998709778:
3214170322: N/A
2924222857:
2998709778:
3 declare initial expr
  1. 1114820451; locally 9835406:
    \(W_{\rm by\ system} = \Delta KE\)
    \(\)
no validation is available for declarations 1114820451:
1114820451:
18 substitute LHS of expr 1 into expr 2
  1. 9510328252; locally 7110498:
    \(KE_{\rm initial} = 0\)
    \(\)
  2. 5779256336; locally 8118190:
    \(W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
  1. 5850144586; locally 2751634:
    \(W_{\rm by\ system} = KE_{\rm final}\)
    \(\)
valid 9510328252:
5779256336:
5850144586:
9510328252:
5779256336:
5850144586:
10 declare initial expr
  1. 8049905441; locally 9781919:
    \(\Delta KE = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
no validation is available for declarations 8049905441:
8049905441:
13 change three variables in expr
  1. 8357234146; locally 5104592:
    \(KE = \frac{1}{2} m v^2\)
    \(\)
  1. 3731774096:
    \(KE\)
    \(\)
  2. 3350802342:
    \(KE_{\rm initial}\)
    \(\)
  3. 5904227750:
    \(m\)
    \(\)
  4. 6281834543:
    \(m_1\)
    \(\)
  5. 8066819515:
    \(v\)
    \(\)
  6. 3274176452:
    \(v_{\rm initial}\)
    \(\)
  1. 6091977310; locally 9031887:
    \(KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2\)
    \(\)
valid 8357234146:
6091977310:
8357234146:
6091977310:
21 substitute LHS of expr 1 into expr 2
  1. 9081138616; locally 6536576:
    \(W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
  2. 2907404069; locally 2619766:
    \(W_{\rm by\ system} = W_{\rm to\ system}\)
    \(\)
  1. 4947831649; locally 8655239:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}\)
    \(\)
valid 9081138616:
2907404069:
4947831649:
9081138616:
2907404069:
4947831649:
20 declare initial expr
  1. 2907404069; locally 2619766:
    \(W_{\rm by\ system} = W_{\rm to\ system}\)
    \(\)
no validation is available for declarations 2907404069:
2907404069:
22 substitute LHS of expr 1 into expr 2
  1. 4393670960; locally 4947999:
    \(W_{\rm to\ system} = \frac{G m_1 m_2}{r}\)
    \(\)
  2. 4947831649; locally 8655239:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}\)
    \(\)
  1. 6892595652; locally 2942416:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}\)
    \(\)
valid 4393670960:
4947831649:
6892595652:
4393670960:
4947831649:
6892595652:
17 substitute LHS of expr 1 into expr 2
  1. 2998709778; locally 6923850:
    \(v_{\rm initial} = 0\)
    \(\)
  2. 6091977310; locally 9031887:
    \(KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2\)
    \(\)
  1. 9510328252; locally 7110498:
    \(KE_{\rm initial} = 0\)
    \(\)
valid 2998709778:
6091977310:
9510328252:
2998709778:
6091977310:
9510328252:
14 change three variables in expr
  1. 8357234146; locally 5104592:
    \(KE = \frac{1}{2} m v^2\)
    \(\)
  1. 4587046017:
    \(KE\)
    \(\)
  2. 3939572542:
    \(KE_{\rm final}\)
    \(\)
  3. 9350720370:
    \(m\)
    \(\)
  4. 3166466250:
    \(m_1\)
    \(\)
  5. 6038673136:
    \(v\)
    \(\)
  6. 1616666229:
    \(v_{\rm final}\)
    \(\)
  1. 8552710882; locally 1397156:
    \(KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
failed 8357234146:
8552710882:
8357234146:
8552710882:
4 declare initial expr
  1. 7882872592; locally 6798426:
    \(W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}\)
    \(\)
no validation is available for declarations 7882872592:
7882872592:
24 square root both sides
  1. 7112646057; locally 4594601:
    \(v_{\rm final}^2 = \frac{2 G m_2}{r}\)
    \(\)
  1. 5846639423; locally 7112224:
    \(v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
  2. 5693047217; locally 1366396:
    \(v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}\)
    \(\)
no check performed 7112646057:
5846639423:
5693047217:
7112646057:
5846639423:
5693047217:
19 substitute LHS of expr 1 into expr 2
  1. 8552710882; locally 1397156:
    \(KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
  2. 5850144586; locally 2751634:
    \(W_{\rm by\ system} = KE_{\rm final}\)
    \(\)
  1. 9081138616; locally 6536576:
    \(W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
valid 8552710882:
5850144586:
9081138616:
8552710882:
5850144586:
9081138616:
23 multiply both sides by
  1. 6892595652; locally 2942416:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}\)
    \(\)
  1. 7410526982:
    \(2/m_1\)
    \(\)
  1. 7112646057; locally 4594601:
    \(v_{\rm final}^2 = \frac{2 G m_2}{r}\)
    \(\)
valid 6892595652:
7112646057:
6892595652:
7112646057:
1 declare initial expr
  1. 3214170322; locally 8462685:
    \(v(r=\infty) = 0\)
    \(\)
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
6777 variable \vec{F}
\(\vec{F}\)		 real dimensionless force 1
6191 variable W_{\rm by\ system}
\(W_{\rm by\ system}\)		 real
  • length: 2
  • mass: 1
  • time: -2
 work done by system 5
4851 variable m_2
\(m_2\)		 real
  • mass: 1
 mass 31
4929 variable KE
\(KE\)		 ['real']
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 7
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
4121 variable KE_{\rm initial}
\(KE_{\rm initial}\)		 real dimensionless initial kinetic energy 5
4202 variable F
\(F\)		 ['real']
  • length: 1
  • mass: 1
  • time: -2
 force 21
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 60
5340 variable KE_{\rm final}
\(KE_{\rm final}\)		 real dimensionless final kinetic energy 5
8909 variable v_{\rm final}
\(v_{\rm final}\)		 real
  • length: 1
  • time: -1
 final velocity 9
5022 variable m_1
\(m_1\)		 real
  • mass: 1
 mass 35
1934 variable v_{\rm initial}
\(v_{\rm initial}\)		 real
  • length: 1
  • time: -1
 initial velocity 4
5734 variable \Delta KE
\(\Delta KE\)		 real dimensionless change in kinetic energy 2
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
2530 variable r
\(r\)		 ['real']
  • length: 1
 radius 60
9372 variable W_{\rm to\ system}
\(W_{\rm to\ system}\)		 real
  • length: 2
  • mass: 1
  • time: -2
 work done to system 8
MESSAGES: