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Review escape velocity

step inference rule input feed output step validity (as per SymPy)
1
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 6935745841
    \(F=G \frac{m_1 m_2}{x^2}\)
 no validation is available for declarations
2
  • 111556: 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}.
  1. 1590774089
    \(dW=F dx\)
  1. 6935745841
    \(F=G \frac{m_1 m_2}{x^2}\)
  1. 8604483515
    \(dW=G \frac{m_1 m_2}{x^2} dx\)
 LHS diff is pdg0004202 - pdg0009398 RHS diff is pdg0004851*pdg0005022*pdg0006277*(1 - pdg0009199)/pdg0004037**2
3
  • 111408: integrate
  • number of inputs: 1; feeds: 0; outputs: 1
  • Integrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}.
  1. 8604483515
    \(dW=G \frac{m_1 m_2}{x^2} dx\)
  1. 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
  • 111662: evaluate definite integral
  • number of inputs: 1; feeds: 0; outputs: 1
  • Evaluate definite integral Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 4447113478
    \(\int dW=G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx\)
  1. 5732331610
    \(W=G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
 Type Tuple cannot be instantiated; use tuple() instead
5
  • 111984: 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}.
  1. 5732331610
    \(W=G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
  1. 1413137236
    \(m_1\)
  1. 9072369552
    \(m_{\rm Earth}\)
  1. 7140470627
    \(m\)
  1. 2764966428
    \(m_2\)
  1. 6131764194
    \(W=G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
 invalid syntax (<string>, line 0)
6
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 6131764194
    \(W=G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
  1. 5978756813
    \(W=G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)
 LHS diff is W - pdg0006789 RHS diff is pdg0005156*pdg0005458*pdg0006277*(pdg0003236 - pdg0004037**2)/(pdg0003236*pdg0004037**2)
7
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 5978756813
    \(W=G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)
  1. 7749253510
    \(W=G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)
 valid
8
  • 111732: 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}.
  1. 7875206161
    \(E_2=KE_2 + PE_2\)
  1. 8558338742
    \(E_2=E_1\)
  1. 4303372136
    \(E_1=KE_1 + PE_1\)
  1. 8960645192
    \(KE_2 + PE_2=KE_1 + PE_1\)
 recognized infrule but not yet supported
9
  • 111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an assumption.
  1. 2267521164
    \(PE_2=0\)
 no validation is available for declarations
10
  • 111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an assumption.
  1. 1840080113
    \(KE_2=0\)
 no validation is available for declarations
11
  • 111732: 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}.
  1. 1840080113
    \(KE_2=0\)
  1. 8960645192
    \(KE_2 + PE_2=KE_1 + PE_1\)
  1. 2267521164
    \(PE_2=0\)
  1. 9749777192
    \(0=KE_1 + PE_1\)
 recognized infrule but not yet supported
12
  • 111984: 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}.
  1. 9749777192
    \(0=KE_1 + PE_1\)
  1. 8416464049
    \(KE_{\rm escape}\)
  1. 5591692598
    \(KE_1\)
  1. 8871333437
    \(PE_{\rm Earth\ surface}\)
  1. 6158970683
    \(PE_1\)
  1. 2503972039
    \(0=KE_{\rm escape} + PE_{\rm Earth\ surface}\)
 list index out of range
13
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 7573835180
    \(PE_{\rm Earth\ surface}=-W\)
 no validation is available for declarations
14
  • 111556: 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}.
  1. 7573835180
    \(PE_{\rm Earth\ surface}=-W\)
  1. 7749253510
    \(W=G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)
  1. 3846041519
    \(PE_{\rm Earth\ surface}=-G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
 LHS diff is -pdg0006431 + pdg0006789 RHS diff is 2*pdg0005156*pdg0005458*pdg0006277/pdg0003236
15
  • 111984: 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}.
  1. 8357234146
    \(KE=\frac{1}{2} m v^2\)
  1. 6681646197
    \(v\)
  1. 6498985149
    \(v_{\rm escape}\)
  1. 9370882921
    \(KE_{\rm escape}\)
  1. 5021965469
    \(KE\)
  1. 6870322215
    \(KE_{\rm escape}=\frac{1}{2} m v_{\rm escape}^2\)
 list index out of range
16
  • 111732: 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}.
  1. 3846041519
    \(PE_{\rm Earth\ surface}=-G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
  1. 2503972039
    \(0=KE_{\rm escape} + PE_{\rm Earth\ surface}\)
  1. 6870322215
    \(KE_{\rm escape}=\frac{1}{2} m v_{\rm escape}^2\)
  1. 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
  • 111530: 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}.
  1. 2042298788
    \(0=-G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2\)
  1. 5050429607
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
  1. 9703482302
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}=\frac{1}{2} m v_{\rm escape}^2\)
 valid
18
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 9703482302
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}=\frac{1}{2} m v_{\rm escape}^2\)
  1. 1143343287
    \(G \frac{m_{\rm Earth}}{r_{\rm Earth}}=\frac{1}{2} v_{\rm escape}^2\)
 LHS diff is pdg0005458*pdg0006277*(pdg0005156 - 1)/pdg0003236 RHS diff is pdg0008656**2*(pdg0005156 - 1)/2
19
  • 111182: 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}.
  1. 1143343287
    \(G \frac{m_{\rm Earth}}{r_{\rm Earth}}=\frac{1}{2} v_{\rm escape}^2\)
  1. 5775658332
    \(2\)
  1. 2977457786
    \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}=v_{\rm escape}^2\)
 valid
20
  • 111268: 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}.
  1. 2977457786
    \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}=v_{\rm escape}^2\)
  1. 9412953728
    \(v_{\rm escape}^2=2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)
 valid
21
  • 111524: 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}.
  1. 9412953728
    \(v_{\rm escape}^2=2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)
  1. 2750380042
    \(v_{\rm escape}=-\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
  1. 1330874553
    \(v_{\rm escape}=\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
 recognized infrule but not yet supported
22
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 1330874553
    \(v_{\rm escape}=\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
 no validation is available for declarations
23
  • 111984: 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}.
  1. 1330874553
    \(v_{\rm escape}=\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
  1. 2135482543
    \(m\)
  1. 8020058613
    \(r\)
  1. 2396787389
    \(r_{\rm Earth}\)
  1. 2674546234
    \(m_{\rm Earth}\)
  1. 5404822208
    \(v_{\rm escape}=\sqrt{2 G \frac{m}{r}}\)
 list index out of range


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