Review Derivation: Physics Derivation Graph

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Review escape velocity

abstract:
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6935745841 $F = G \frac{m_1 m_2}{x^2}$	no validation is available for declarations
2	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1590774089 $dW = F dx$ 6935745841 $F = G \frac{m_1 m_2}{x^2}$		8604483515 $dW = G \frac{m_1 m_2}{x^2} dx$	LHS diff is pdg0004202 - pdg0009398 RHS diff is pdg0004851pdg0005022pdg0006277(1 - pdg0009199)/pdg0004037*2
3	0000111408: integrate number of inputs: 1; feeds: 0; outputs: 1 Integrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}.	8604483515 $dW = G \frac{m_1 m_2}{x^2} dx$		4447113478 $\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx$	recognized infrule but not yet supported
4	0000111662: evaluate definite integral number of inputs: 1; feeds: 0; outputs: 1 Evaluate definite integral Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	4447113478 $\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx$		5732331610 $W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$	Not evaluated due to missing term in SymPy
5	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	5732331610 $W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$	7140470627 $m$ 2764966428 $m_2$ 9072369552 $m_{\rm Earth}$ 1413137236 $m_1$	6131764194 $W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$	Not evaluated due to missing term in SymPy
6	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	6131764194 $W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$		5978756813 $W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)$	LHS diff is W - pdg0006789 RHS diff is pdg0005156pdg0005458pdg0006277(pdg0003236 - pdg00040372)/(pdg0003236pdg0004037**2)
7	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	5978756813 $W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)$		7749253510 $W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}$	valid
8	0000111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	8558338742 $E_2 = E_1$ 7875206161 $E_2 = KE_2 + PE_2$ 4303372136 $E_1 = KE_1 + PE_1$		8960645192 $KE_2 + PE_2 = KE_1 + PE_1$	recognized infrule but not yet supported
9	0000111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an assumption.			2267521164 $PE_2 = 0$	no validation is available for declarations
10	0000111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an assumption.			1840080113 $KE_2 = 0$	no validation is available for declarations
11	0000111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	8960645192 $KE_2 + PE_2 = KE_1 + PE_1$ 1840080113 $KE_2 = 0$ 2267521164 $PE_2 = 0$		9749777192 $0 = KE_1 + PE_1$	recognized infrule but not yet supported
12	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	9749777192 $0 = KE_1 + PE_1$	8871333437 $PE_{\rm Earth\ surface}$ 6158970683 $PE_1$ 8416464049 $KE_{\rm escape}$ 5591692598 $KE_1$	2503972039 $0 = KE_{\rm escape} + PE_{\rm Earth\ surface}$	LHS diff is 0 RHS diff is pdg0001955 + pdg0004093 - pdg0005332 - pdg0006431
13	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			7573835180 $PE_{\rm Earth\ surface} = -W$	no validation is available for declarations
14	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	7573835180 $PE_{\rm Earth\ surface} = -W$ 7749253510 $W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}$		3846041519 $PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}$	LHS diff is -pdg0006431 + pdg0006789 RHS diff is 2pdg0005156pdg0005458*pdg0006277/pdg0003236
15	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	8357234146 $KE = \frac{1}{2} m v^2$	6498985149 $v_{\rm escape}$ 6681646197 $v$ 9370882921 $KE_{\rm escape}$ 5021965469 $KE$	6870322215 $KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2$	LHS diff is pdg0004929 - pdg0005332 RHS diff is pdg0005156(pdg00013572 - pdg0008656*2)/2
16	0000111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	2503972039 $0 = KE_{\rm escape} + PE_{\rm Earth\ surface}$ 3846041519 $PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}$ 6870322215 $KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2$		2042298788 $0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2$	recognized infrule but not yet supported
17	0000111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2042298788 $0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2$	5050429607 $G \frac{m_{\rm Earth} m}{r_{\rm Earth}}$	9703482302 $G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2$	valid
18	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9703482302 $G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2$		1143343287 $G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2$	LHS diff is pdg0005458pdg0006277(pdg0005156 - 1)/pdg0003236 RHS diff is pdg0008656*2(pdg0005156 - 1)/2
19	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	1143343287 $G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2$	5775658332 $2$	2977457786 $2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2$	valid
20	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	2977457786 $2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2$		9412953728 $v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}$	valid
21	0000111524: square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.	9412953728 $v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}$		2750380042 $v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$ 1330874553 $v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$
22	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	1330874553 $v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$			no validation is available for declarations
Note about step 23: replaced Earth-specific variables
23	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	1330874553 $v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$	8020058613 $r$ 2396787389 $r_{\rm Earth}$ 2135482543 $m$ 2674546234 $m_{\rm Earth}$	5404822208 $v_{\rm escape} = \sqrt{2 G \frac{m}{r}}$	LHS diff is 0 RHS diff is sqrt(2)(-sqrt(pdg0005156pdg0006277/pdg0002530) + sqrt(pdg0005458*pdg0006277/pdg0003236))

Symbols used in escape velocity

0000009398: $ dW $
0000009199: $ dx $
0000005579: $ E_1 $
0000004550: $ E_2 $
0000004202: $ F $
0000006277: $ G $
0000001552: $ j $
0000004929: $ KE $
0000001955: $ KE_1 $
0000001352: $ KE_2 $
0000005332: $ KE_{\rm escape} $
0000005156: $ m $
0000005022: $ m_1 $
0000004851: $ m_2 $
0000005458: $ m_{\rm Earth} $
0000004093: $ PE_1 $
0000008849: $ PE_2 $
0000006431: $ PE_{\rm Earth\ surface} $
0000002530: $ r $
0000003236: $ r_{\rm Earth} $
0000001357: $ v $
0000008656: $ v_{\rm escape} $
0000006789: $ W $
0000004037: $ x $

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0.007 seconds for pdg_app/to_review_derivation a9c7a62c-0543-4066-8db7-6b9225d45550
0.014 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationbb597d6a-aaba-4e63-8a43-722acb4e78b5
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0.008 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDbb597d6a-aaba-4e63-8a43-722acb4e78b5
0.006 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTbb597d6a-aaba-4e63-8a43-722acb4e78b5
0.006 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepbb597d6a-aaba-4e63-8a43-722acb4e78b5
0.023 seconds for pdg_app/ a9c7a62c-0543-4066-8db7-6b9225d45550

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6935745841 \(F = G \frac{m_1 m_2}{x^2}\)	no validation is available for declarations
2	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1590774089 \(dW = F dx\) 6935745841 \(F = G \frac{m_1 m_2}{x^2}\)		8604483515 \(dW = G \frac{m_1 m_2}{x^2} dx\)	LHS diff is pdg0004202 - pdg0009398 RHS diff is pdg0004851pdg0005022pdg0006277(1 - pdg0009199)/pdg0004037*2
3	0000111408: integrate number of inputs: 1; feeds: 0; outputs: 1 Integrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}.	8604483515 \(dW = G \frac{m_1 m_2}{x^2} dx\)		4447113478 \(\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx\)	recognized infrule but not yet supported
4	0000111662: evaluate definite integral number of inputs: 1; feeds: 0; outputs: 1 Evaluate definite integral Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	4447113478 \(\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx\)		5732331610 \(W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)	Not evaluated due to missing term in SymPy
5	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	5732331610 \(W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)	7140470627 \(m\) 2764966428 \(m_2\) 9072369552 \(m_{\rm Earth}\) 1413137236 \(m_1\)	6131764194 \(W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)	Not evaluated due to missing term in SymPy
6	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	6131764194 \(W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)		5978756813 \(W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)	LHS diff is W - pdg0006789 RHS diff is pdg0005156pdg0005458pdg0006277(pdg0003236 - pdg00040372)/(pdg0003236pdg0004037**2)
7	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	5978756813 \(W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)		7749253510 \(W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)	valid
8	0000111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	8558338742 \(E_2 = E_1\) 7875206161 \(E_2 = KE_2 + PE_2\) 4303372136 \(E_1 = KE_1 + PE_1\)		8960645192 \(KE_2 + PE_2 = KE_1 + PE_1\)	recognized infrule but not yet supported
9	0000111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an assumption.			2267521164 \(PE_2 = 0\)	no validation is available for declarations
10	0000111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an assumption.			1840080113 \(KE_2 = 0\)	no validation is available for declarations
11	0000111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	8960645192 \(KE_2 + PE_2 = KE_1 + PE_1\) 1840080113 \(KE_2 = 0\) 2267521164 \(PE_2 = 0\)		9749777192 \(0 = KE_1 + PE_1\)	recognized infrule but not yet supported
12	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	9749777192 \(0 = KE_1 + PE_1\)	8871333437 \(PE_{\rm Earth\ surface}\) 6158970683 \(PE_1\) 8416464049 \(KE_{\rm escape}\) 5591692598 \(KE_1\)	2503972039 \(0 = KE_{\rm escape} + PE_{\rm Earth\ surface}\)	LHS diff is 0 RHS diff is pdg0001955 + pdg0004093 - pdg0005332 - pdg0006431
13	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			7573835180 \(PE_{\rm Earth\ surface} = -W\)	no validation is available for declarations
14	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	7573835180 \(PE_{\rm Earth\ surface} = -W\) 7749253510 \(W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)		3846041519 \(PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)	LHS diff is -pdg0006431 + pdg0006789 RHS diff is 2pdg0005156pdg0005458*pdg0006277/pdg0003236
15	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	8357234146 \(KE = \frac{1}{2} m v^2\)	6498985149 \(v_{\rm escape}\) 6681646197 \(v\) 9370882921 \(KE_{\rm escape}\) 5021965469 \(KE\)	6870322215 \(KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2\)	LHS diff is pdg0004929 - pdg0005332 RHS diff is pdg0005156(pdg00013572 - pdg0008656*2)/2
16	0000111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	2503972039 \(0 = KE_{\rm escape} + PE_{\rm Earth\ surface}\) 3846041519 \(PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\) 6870322215 \(KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2\)		2042298788 \(0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2\)	recognized infrule but not yet supported
17	0000111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2042298788 \(0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2\)	5050429607 \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)	9703482302 \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2\)	valid
18	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9703482302 \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2\)		1143343287 \(G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2\)	LHS diff is pdg0005458pdg0006277(pdg0005156 - 1)/pdg0003236 RHS diff is pdg0008656*2(pdg0005156 - 1)/2
19	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	1143343287 \(G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2\)	5775658332 \(2\)	2977457786 \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2\)	valid
20	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	2977457786 \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2\)		9412953728 \(v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)	valid
21	0000111524: square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.	9412953728 \(v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)		2750380042 \(v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\) 1330874553 \(v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
22	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	1330874553 \(v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)			no validation is available for declarations
Note about step 23: replaced Earth-specific variables
23	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	1330874553 \(v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)	8020058613 \(r\) 2396787389 \(r_{\rm Earth}\) 2135482543 \(m\) 2674546234 \(m_{\rm Earth}\)	5404822208 \(v_{\rm escape} = \sqrt{2 G \frac{m}{r}}\)	LHS diff is 0 RHS diff is sqrt(2)(-sqrt(pdg0005156pdg0006277/pdg0002530) + sqrt(pdg0005458*pdg0006277/pdg0003236))