## 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

symbol ID category latex scope dimension name value Used in derivations references
4121 variable KE_{\rm initial}
$$KE_{\rm initial}$$
real dimensionless initial kinetic energy
5
8909 variable v_{\rm final}
$$v_{\rm final}$$
real
• length: 1
• time: -1
final velocity
9
4037 variable x
$$x$$
['real']
• length: 1
position
53
5156 variable m
$$m$$
['real']
• mass: 1
mass
69
1357 variable v
$$v$$
['real']
• length: 1
• time: -1
velocity
83
4929 variable KE
$$KE$$
['real']
• length: 2
• mass: 1
• time: -2
kinetic energy
7
5340 variable KE_{\rm final}
$$KE_{\rm final}$$
real dimensionless final kinetic energy
5
2530 variable r
$$r$$
['real']
• length: 1
60
6777 variable \vec{F}
$$\vec{F}$$
real dimensionless force
1
5734 variable \Delta KE
$$\Delta KE$$
real dimensionless change in kinetic energy
2
1934 variable v_{\rm initial}
$$v_{\rm initial}$$
real
• length: 1
• time: -1
initial velocity
4
5022 variable m_1
$$m_1$$
real
• mass: 1
mass
35
9372 variable W_{\rm to\ system}
$$W_{\rm to\ system}$$
real
• length: 2
• mass: 1
• time: -2
work done to system
8
4202 variable F
$$F$$
['real']
• length: 1
• mass: 1
• time: -2
force
21
6191 variable W_{\rm by\ system}
$$W_{\rm by\ system}$$
real
• length: 2
• mass: 1
• time: -2
work done by system
5
6277 constant G
$$G$$
real
• length: 3
• mass: -1
• time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
60
4851 variable m_2
$$m_2$$
real
• mass: 1
mass
31
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 1153771: unable to eval AST for "Equality(Symbol('pdg9372'), Integral(Dot(Symbol('pdg6777'), Symbol('pdg2530')), Tuple(oo, Symbol('pdg2530'))))" aka "sympy.Equality(sympy.Symbol('pdg9372'), sympy.Integral(Dot(sympy.Symbol('pdg6777'), sympy.Symbol('pdg2530')), sympy.Tuple(oo, sympy.Symbol('pdg2530'))))"
• in step 1153771: unable to eval AST for "Equality(Symbol('pdg9372'), Integral(Dot(Symbol('pdg6777'), Symbol('pdg2530')), Tuple(oo, Symbol('pdg2530'))))" aka "sympy.Equality(sympy.Symbol('pdg9372'), sympy.Integral(Dot(sympy.Symbol('pdg6777'), sympy.Symbol('pdg2530')), sympy.Tuple(oo, sympy.Symbol('pdg2530'))))"
• in step 7925705: unable to eval AST for "Symbol('pdg4929'))" aka "sympy.Symbol('pdg4929'))"
• in step 7936249: unable to eval AST for "Equality(Symbol('pdg9372'), Integral(Dot(Symbol('pdg6777'), Symbol('pdg2530')), Tuple(oo, Symbol('pdg2530'))))" aka "sympy.Equality(sympy.Symbol('pdg9372'), sympy.Integral(Dot(sympy.Symbol('pdg6777'), sympy.Symbol('pdg2530')), sympy.Tuple(oo, sympy.Symbol('pdg2530'))))"
• unable to eval AST for "Equality(Symbol('pdg9372'), Integral(Dot(Symbol('pdg6777'), Symbol('pdg2530')), Tuple(oo, Symbol('pdg2530'))))" aka "sympy.Equality(sympy.Symbol('pdg9372'), sympy.Integral(Dot(sympy.Symbol('pdg6777'), sympy.Symbol('pdg2530')), sympy.Tuple(oo, sympy.Symbol('pdg2530'))))"