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

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

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.

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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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Physics Derivation Graph: Steps for velocity at distance r of object dropped from infinity
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	5902985919; locally 3470082: $\vec{F} = G \frac{m_{1} m_{2}}{x^{2}} \hat{x}$ 7882872592; locally 6798426: $W_{t o s y s t e m} = \int_{\infty}^{r} \vec{F} \cdot d \vec{r}$		3566149658; locally 7300369: $W_{t o s y s t e m} = \int_{\infty}^{r} \frac{- G m_{1} m_{2}}{x^{2}} d x$	failed	5902985919: 7882872592: 3566149658:	5902985919: 7882872592: 3566149658:
2	declare initial expr			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	8049905441; locally 9781919: $Δ K E = K E_{f i n a l} - K E_{i n i t i a l}$ 1114820451; locally 9835406: $W_{b y s y s t e m} = Δ K E$		5779256336; locally 8118190: $W_{b y s y s t e m} = K E_{f i n a l} - K E_{i n i t i a l}$	valid	8049905441: 1114820451: 5779256336:	8049905441: 1114820451: 5779256336:
15	declare initial expr			2924222857; locally 1712972: $v_{i n i t i a l} = v (r = \infty)$	no validation is available for declarations	2924222857:	2924222857:
8	simplify	5596822289; locally 5818573: $W_{t o s y s t e m} = - G m_{1} m_{2} ({\frac{- 1}{x} \|}_{\infty}^{r})$		2061086175; locally 2429271: $W_{t o s y s t e m} = - G m_{1} m_{2} (\frac{- 1}{r} - \frac{- 1}{\infty})$	LHS diff is 0 RHS diff is pdg5022pdg6277(-pdg4851 + pdg4851(-1/pdg2530))	5596822289: 2061086175:	5596822289: 2061086175:
11	declare initial expr			8357234146; locally 5104592: $K E = \frac{1}{2} m v^{2}$	no validation is available for declarations	8357234146:	8357234146:
26	declare final expr	2005061870; locally 3435796: $v (r) = \sqrt{\frac{2 G m_{2}}{r}}$			no validation is available for declarations	2005061870:	2005061870:
7	evaluate definite integral	8405272745; locally 9707318: $W_{t o s y s t e m} = - G m_{1} m_{2} \int_{\infty}^{r} \frac{1}{x^{2}} d x$		5596822289; locally 5818573: $W_{t o s y s t e m} = - G m_{1} m_{2} ({\frac{- 1}{x} \|}_{\infty}^{r})$	LHS diff is 0 RHS diff is pdg4851pdg5022pdg6277*(1 + 1/pdg2530)	8405272745: 5596822289:	8405272745: 5596822289:
9	simplify	2061086175; locally 2429271: $W_{t o s y s t e m} = - G m_{1} m_{2} (\frac{- 1}{r} - \frac{- 1}{\infty})$		4393670960; locally 4947999: $W_{t o s y s t e m} = \frac{G m_{1} m_{2}}{r}$	LHS diff is 0 RHS diff is pdg5022pdg6277(-pdg2530*pdg4851(-1/pdg2530) - pdg4851)/pdg2530	2061086175: 4393670960:	2061086175: 4393670960:
25	change variable X to Y	5846639423; locally 7112224: $v_{f i n a l} = \sqrt{\frac{2 G m_{2}}{r}}$	6599829782: $v_{f i n a l}$ 3531380618: $v (r)$	2005061870; locally 3435796: $v (r) = \sqrt{\frac{2 G m_{2}}{r}}$	valid	5846639423: 2005061870:	5846639423: 2005061870:
6	simplify	3566149658; locally 7300369: $W_{t o s y s t e m} = \int_{\infty}^{r} \frac{- G m_{1} m_{2}}{x^{2}} d x$		8405272745; locally 9707318: $W_{t o s y s t e m} = - G m_{1} m_{2} \int_{\infty}^{r} \frac{1}{x^{2}} d x$	valid	3566149658: 8405272745:	3566149658: 8405272745:
16	substitute LHS of expr 1 into expr 2	3214170322; locally 8462685: $v (r = \infty) = 0$ 2924222857; locally 1712972: $v_{i n i t i a l} = v (r = \infty)$		2998709778; locally 6923850: $v_{i n i t i a l} = 0$	Nothing to split	3214170322: no LHS/RHS split 2924222857: 2998709778:	3214170322: N/A 2924222857: 2998709778:
3	declare initial expr			1114820451; locally 9835406: $W_{b y s y s t e m} = Δ K E$	no validation is available for declarations	1114820451:	1114820451:
18	substitute LHS of expr 1 into expr 2	9510328252; locally 7110498: $K E_{i n i t i a l} = 0$ 5779256336; locally 8118190: $W_{b y s y s t e m} = K E_{f i n a l} - K E_{i n i t i a l}$		5850144586; locally 2751634: $W_{b y s y s t e m} = K E_{f i n a l}$	valid	9510328252: 5779256336: 5850144586:	9510328252: 5779256336: 5850144586:
10	declare initial expr			8049905441; locally 9781919: $Δ K E = K E_{f i n a l} - K E_{i n i t i a l}$	no validation is available for declarations	8049905441:	8049905441:
13	change three variables in expr	8357234146; locally 5104592: $K E = \frac{1}{2} m v^{2}$	3731774096: $K E$ 3350802342: $K E_{i n i t i a l}$ 5904227750: $m$ 6281834543: $m_{1}$ 8066819515: $v$ 3274176452: $v_{i n i t i a l}$	6091977310; locally 9031887: $K E_{i n i t i a l} = \frac{1}{2} m_{1} v_{i n i t i a l}^{2}$	valid	8357234146: 6091977310:	8357234146: 6091977310:
21	substitute LHS of expr 1 into expr 2	9081138616; locally 6536576: $W_{b y s y s t e m} = \frac{1}{2} m_{1} v_{f i n a l}^{2}$ 2907404069; locally 2619766: $W_{b y s y s t e m} = W_{t o s y s t e m}$		4947831649; locally 8655239: $\frac{1}{2} m_{1} v_{f i n a l}^{2} = W_{t o s y s t e m}$	valid	9081138616: 2907404069: 4947831649:	9081138616: 2907404069: 4947831649:
20	declare initial expr			2907404069; locally 2619766: $W_{b y s y s t e m} = W_{t o s y s t e m}$	no validation is available for declarations	2907404069:	2907404069:
22	substitute LHS of expr 1 into expr 2	4393670960; locally 4947999: $W_{t o s y s t e m} = \frac{G m_{1} m_{2}}{r}$ 4947831649; locally 8655239: $\frac{1}{2} m_{1} v_{f i n a l}^{2} = W_{t o s y s t e m}$		6892595652; locally 2942416: $\frac{1}{2} m_{1} v_{f i n a l}^{2} = \frac{G m_{1} m_{2}}{r}$	valid	4393670960: 4947831649: 6892595652:	4393670960: 4947831649: 6892595652:
17	substitute LHS of expr 1 into expr 2	2998709778; locally 6923850: $v_{i n i t i a l} = 0$ 6091977310; locally 9031887: $K E_{i n i t i a l} = \frac{1}{2} m_{1} v_{i n i t i a l}^{2}$		9510328252; locally 7110498: $K E_{i n i t i a l} = 0$	valid	2998709778: 6091977310: 9510328252:	2998709778: 6091977310: 9510328252:
14	change three variables in expr	8357234146; locally 5104592: $K E = \frac{1}{2} m v^{2}$	4587046017: $K E$ 3939572542: $K E_{f i n a l}$ 9350720370: $m$ 3166466250: $m_{1}$ 6038673136: $v$ 1616666229: $v_{f i n a l}$	8552710882; locally 1397156: $K E_{f i n a l} = \frac{1}{2} m_{1} v_{f i n a l}^{2}$	failed	8357234146: 8552710882:	8357234146: 8552710882:
4	declare initial expr			7882872592; locally 6798426: $W_{t o s y s t e m} = \int_{\infty}^{r} \vec{F} \cdot d \vec{r}$	no validation is available for declarations	7882872592:	7882872592:
24	square root both sides	7112646057; locally 4594601: $v_{f i n a l}^{2} = \frac{2 G m_{2}}{r}$		5846639423; locally 7112224: $v_{f i n a l} = \sqrt{\frac{2 G m_{2}}{r}}$ 5693047217; locally 1366396: $v_{f i n a l} = - \sqrt{\frac{2 G m_{2}}{r}}$	no check performed	7112646057: 5846639423: 5693047217:	7112646057: 5846639423: 5693047217:
19	substitute LHS of expr 1 into expr 2	8552710882; locally 1397156: $K E_{f i n a l} = \frac{1}{2} m_{1} v_{f i n a l}^{2}$ 5850144586; locally 2751634: $W_{b y s y s t e m} = K E_{f i n a l}$		9081138616; locally 6536576: $W_{b y s y s t e m} = \frac{1}{2} m_{1} v_{f i n a l}^{2}$	valid	8552710882: 5850144586: 9081138616:	8552710882: 5850144586: 9081138616:
23	multiply both sides by	6892595652; locally 2942416: $\frac{1}{2} m_{1} v_{f i n a l}^{2} = \frac{G m_{1} m_{2}}{r}$	7410526982: $2 / m_{1}$	7112646057; locally 4594601: $v_{f i n a l}^{2} = \frac{2 G m_{2}}{r}$	valid	6892595652: 7112646057:	6892595652: 7112646057:
1	declare initial expr			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

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 $m_{2}$	real	mass: 1	mass		Kepler's Third Law: period squared propto distance cubed escape velocity Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity	Mass	31
5734	variable	\Delta KE $Δ K E$	real	dimensionless	change in kinetic energy		velocity at distance r of object dropped from infinity	Kinetic_energy	2
6191	variable	W_{\rm by\ system} $W_{b y s y s t e m}$	real	length: 2 mass: 1 time: -2	work done by system		velocity at distance r of object dropped from infinity	Work_(physics)	5
4929	variable	KE $K E$	['real']	length: 2 mass: 1 time: -2	kinetic energy		velocity at distance r of object dropped from infinity escape velocity time invariant force conserves energy	Kinetic_energy	7
9372	variable	W_{\rm to\ system} $W_{t o s y s t e m}$	real	length: 2 mass: 1 time: -2	work done to system		velocity at distance r of object dropped from infinity	Work_(physics)	8
1934	variable	v_{\rm initial} $v_{i n i t i a l}$	real	length: 1 time: -1	initial velocity		velocity at distance r of object dropped from infinity	Velocity	4
8909	variable	v_{\rm final} $v_{f i n a l}$	real	length: 1 time: -1	final velocity		velocity at distance r of object dropped from infinity	Velocity	9
1357	variable	v $v$	['real']	length: 1 time: -1	velocity		frequency relations work and force and energy escape velocity Newton's Law of Gravitation derivation of Schrodinger Equation radius for satellite in geostationary orbit equations of motion in 1D with constant acceleration - SUVAT (algebra) speed of Earth around Sun velocity at distance r of object dropped from infinity time invariant force conserves energy Lorentz transformation	Velocity	83
5156	variable	m $m$	['real']	mass: 1	mass		Kepler's Third Law: period squared propto distance cubed equation of motion for a spring escape velocity Schwarzschild radius for non-rotating black hole Newton's Law of Gravitation work and force and energy derivation of Schrodinger Equation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity time invariant force conserves energy	Mass Q11423 Mass	69
5022	variable	m_1 $m_{1}$	real	mass: 1	mass		Kepler's Third Law: period squared propto distance cubed escape velocity Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity	Mass	35
6277	constant	G $G$	real	length: 3 mass: -1 time: -2	gravitational constant	6.6743010^{-11} m^3 kg^-1 * s^-2	Kepler's Third Law: period squared propto distance cubed escape velocity Schwarzschild radius for non-rotating black hole Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity equations of motion in 2D (calculus)	Gravitational_constant	60
4202	variable	F $F$	['real']	length: 1 mass: 1 time: -2	force		escape velocity work and force and energy equation of motion for a spring Newton's Law of Gravitation radius for satellite in geostationary orbit mass of the Earth velocity at distance r of object dropped from infinity time invariant force conserves energy	Force	21
5340	variable	KE_{\rm final} $K E_{f i n a l}$	real	dimensionless	final kinetic energy		velocity at distance r of object dropped from infinity	Kinetic_energy	5
4121	variable	KE_{\rm initial} $K E_{i n i t i a l}$	real	dimensionless	initial kinetic energy		velocity at distance r of object dropped from infinity	Kinetic_energy	5
2530	variable	r $r$	['real']	length: 1	radius		Kepler's Third Law: period squared propto distance cubed escape velocity Schwarzschild radius for non-rotating black hole Newton's Law of Gravitation radius for satellite in geostationary orbit speed of Earth around Sun velocity at distance r of object dropped from infinity	Radius	60
4037	variable	x $x$	['real']	length: 1	position		double intensity when phase is coherent (optics) equation of motion for a spring work and force and energy angle of maximum distance for projectile motion escape velocity projectile path in 2D is parabolic radius for satellite in geostationary orbit particle in a 1D box mass of the Earth Euler equation: trig square root velocity at distance r of object dropped from infinity time invariant force conserves energy equations of motion in 2D (calculus) Lorentz transformation	Position_(geometry)	53
6777	variable	\vec{F} $\vec{F}$	real	dimensionless	force		velocity at distance r of object dropped from infinity	Force Q11402	1

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'))))"