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Review velocity at distance r of object dropped from infinity

step inference rule input feed output step validity (as per SymPy)
Note about step 1: starting velocity at infinity is zero
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 3214170322
    \(v(r=\infty) = 0\)
 no validation is available for declarations
Note about step 2: https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form
2
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5902985919
    \(\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}\)
 no validation is available for declarations
3
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 1114820451
    \(W_{\rm by\ system} = \Delta KE\)
 no validation is available for declarations
4
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 7882872592
    \(W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}\)
 no validation is available for declarations
5
  • 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}.
  1. 7882872592
    \(W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}\)
  2. 5902985919
    \(\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}\)
  1. 3566149658
    \(W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
 LHS diff is pdg0004202 - pdg0009372 RHS diff is pdg0004851*pdg0005022*pdg0006277*(1 - pdg0004037/pdg0002530)/pdg0004037
6
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 3566149658
    \(W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
  1. 8405272745
    \(W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
 valid
7
  • 0000111662: evaluate definite integral
  • number of inputs: 1; feeds: 0; outputs: 1
  • Evaluate definite integral Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8405272745
    \(W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
  1. 5596822289
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
 LHS diff is 0 RHS diff is pdg0004851*pdg0005022*pdg0006277*(pdg0002530 + 1)/pdg0002530
8
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 5596822289
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
  1. 2061086175
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
 LHS diff is 0 RHS diff is pdg0005022*pdg0006277*(-pdg0004851 + pdg0004851(-1/pdg0002530))
9
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 2061086175
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
  1. 4393670960
    \(W_{\rm to\ system} = \frac{G m_1 m_2}{r}\)
 LHS diff is 0 RHS diff is pdg0005022*pdg0006277*(-pdg0002530*pdg0004851(-1/pdg0002530) - pdg0004851)/pdg0002530
10
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8049905441
    \(\Delta KE = KE_{\rm final} - KE_{\rm initial}\)
 no validation is available for declarations
11
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8357234146
    \(KE = \frac{1}{2} m v^2\)
 no validation is available for declarations
12
  • 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}.
  1. 1114820451
    \(W_{\rm by\ system} = \Delta KE\)
  2. 8049905441
    \(\Delta KE = KE_{\rm final} - KE_{\rm initial}\)
  1. 5779256336
    \(W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}\)
 LHS diff is pdg0005734 - pdg0006191 RHS diff is 0
13
  • 0000111236: change three variables in expression
  • number of inputs: 1; feeds: 6; outputs: 1
  • Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.
  1. 8357234146
    \(KE = \frac{1}{2} m v^2\)
  1. 3274176452
    \(v_{\rm initial}\)
  2. 8066819515
    \(v\)
  3. 6281834543
    \(m_1\)
  4. 5904227750
    \(m\)
  5. 3350802342
    \(KE_{\rm initial}\)
  6. 3731774096
    \(KE\)
  1. 6091977310
    \(KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2\)
 LHS diff is -pdg0004121 + pdg0004929 RHS diff is pdg0001357**2*pdg0005156/2 - pdg0001934**2*pdg0005022/2
14
  • 0000111236: change three variables in expression
  • number of inputs: 1; feeds: 6; outputs: 1
  • Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.
  1. 8357234146
    \(KE = \frac{1}{2} m v^2\)
  1. 1616666229
    \(v_{\rm final}\)
  2. 6038673136
    \(v\)
  3. 3166466250
    \(m_1\)
  4. 9350720370
    \(m\)
  5. 3939572542
    \(KE_{\rm final}\)
  6. 4587046017
    \(KE\)
  1. 8552710882
    \(KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2\)
 LHS diff is pdg0004929 - pdg0005340 RHS diff is pdg0001357**2*pdg0005156/2 - pdg0005022*pdg0008909**2/2
15
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2924222857
    \(v_{\rm initial} = v(r=\infty)\)
 no validation is available for declarations
16
  • 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}.
  1. 2924222857
    \(v_{\rm initial} = v(r=\infty)\)
  2. 3214170322
    \(v(r=\infty) = 0\)
  1. 2998709778
    \(v_{\rm initial} = 0\)
 Not evaluated due to missing term in SymPy
17
  • 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}.
  1. 6091977310
    \(KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2\)
  2. 2998709778
    \(v_{\rm initial} = 0\)
  1. 9510328252
    \(KE_{\rm initial} = 0\)
 LHS diff is pdg0001934 - pdg0004121 RHS diff is 0
18
  • 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}.
  1. 5779256336
    \(W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}\)
  2. 9510328252
    \(KE_{\rm initial} = 0\)
  1. 5850144586
    \(W_{\rm by\ system} = KE_{\rm final}\)
 LHS diff is pdg0004121 - pdg0006191 RHS diff is -pdg0005340
19
  • 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}.
  1. 5850144586
    \(W_{\rm by\ system} = KE_{\rm final}\)
  2. 8552710882
    \(KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2\)
  1. 9081138616
    \(W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2\)
 LHS diff is pdg0005340 - pdg0006191 RHS diff is 0
20
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2907404069
    \(W_{\rm by\ system} = W_{\rm to\ system}\)
 no validation is available for declarations
21
  • 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}.
  1. 2907404069
    \(W_{\rm by\ system} = W_{\rm to\ system}\)
  2. 9081138616
    \(W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2\)
  1. 4947831649
    \(\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}\)
 LHS diff is -pdg0005022*pdg0008909**2/2 + pdg0009372 RHS diff is pdg0005022*pdg0008909**2/2 - pdg0009372
22
  • 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}.
  1. 4947831649
    \(\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}\)
  2. 4393670960
    \(W_{\rm to\ system} = \frac{G m_1 m_2}{r}\)
  1. 6892595652
    \(\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}\)
 LHS diff is -pdg0005022*pdg0008909**2/2 + pdg0009372 RHS diff is 0
23
  • 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}.
  1. 6892595652
    \(\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}\)
  1. 7410526982
    \(2/m_1\)
  1. 7112646057
    \(v_{\rm final}^2 = \frac{2 G m_2}{r}\)
 valid
24
  • 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}.
  1. 7112646057
    \(v_{\rm final}^2 = \frac{2 G m_2}{r}\)
  1. 5693047217
    \(v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}\)
  2. 5846639423
    \(v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}\)
25
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 5846639423
    \(v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}\)
  1. 3531380618
    \(v(r)\)
  2. 6599829782
    \(v_{\rm final}\)
  1. 2005061870
    \(v(r) = \sqrt{\frac{2 G m_2}{r}}\)
 LHS diff is pdg0008909 - pdg0001357(pdg0002530) RHS diff is 0
26
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 2005061870
    \(v(r) = \sqrt{\frac{2 G m_2}{r}}\)
no validation is available for declarations

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