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Review time invariant force conserves energy

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.
  1. 5136652623
    \(E = KE + PE\)
 no validation is available for declarations
2
  • 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. 5136652623
    \(E = KE + PE\)
  1. 5803210729
    \(PE_2\)
  1. 5075406409
    \(PE\)
  1. 7939947931
    \(KE_2\)
  1. 6383056612
    \(KE\)
  1. 2344320475
    \(E_2\)
  1. 1258245373
    \(E\)
  1. 7875206161
    \(E_2 = KE_2 + PE_2\)
 LHS diff is -pdg0004550 + pdg0004931 RHS diff is -pdg0001352 + pdg0004929 + pdg0004930 - pdg0008849
3
  • 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. 5136652623
    \(E = KE + PE\)
  1. 8061701434
    \(PE_1\)
  1. 3809726424
    \(PE\)
  1. 1092872200
    \(KE_1\)
  1. 4147101187
    \(KE\)
  1. 4213426349
    \(E_1\)
  1. 3749492596
    \(E\)
  1. 4303372136
    \(E_1 = KE_1 + PE_1\)
 LHS diff is pdg0004931 - pdg0005579 RHS diff is -pdg0001955 - pdg0004093 + pdg0004929 + pdg0004930
5
  • 0000111222: subtract expr 1 from expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Subtract Eq.~\ref{eq:#1} from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.
  1. 7875206161
    \(E_2 = KE_2 + PE_2\)
  1. 4303372136
    \(E_1 = KE_1 + PE_1\)
  1. 5514556106
    \(E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)\)
 LHS diff is -2*pdg0004550 + 2*pdg0005579 RHS diff is -2*pdg0001352 + 2*pdg0001955 + 2*pdg0004093 - 2*pdg0008849
6
  • 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
7
  • 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}.
  1. 8357234146
    \(KE = \frac{1}{2} m v^2\)
  1. 7408927653
    \(KE\)
  1. 5011888122
    \(v_2\)
  1. 9305761407
    \(v\)
  1. 6838659900
    \(KE_2\)
  1. 7676652285
    \(KE_2 = \frac{1}{2} m v_2^2\)
 LHS diff is -pdg0001352 + pdg0004929 RHS diff is pdg0005156*(pdg0001352**2 - pdg0004770**2)/2
8
  • 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}.
  1. 8357234146
    \(KE = \frac{1}{2} m v^2\)
  1. 1934877404
    \(KE\)
  1. 3105350101
    \(v_1\)
  1. 5398681503
    \(v\)
  1. 6964468708
    \(KE_1\)
  1. 4928007622
    \(KE_1 = \frac{1}{2} m v_1^2\)
 LHS diff is -pdg0001955 + pdg0004929 RHS diff is pdg0005156*(pdg0001955**2 - pdg0002473**2)/2
10
  • 0000111222: subtract expr 1 from expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Subtract Eq.~\ref{eq:#1} from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.
  1. 7676652285
    \(KE_2 = \frac{1}{2} m v_2^2\)
  1. 4928007622
    \(KE_1 = \frac{1}{2} m v_1^2\)
  1. 5733146966
    \(KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)\)
 LHS diff is -2*pdg0001352 + 2*pdg0001955 RHS diff is pdg0005156*(pdg0002473**2 - pdg0004770**2)
11
  • 0000111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 5733146966
    \(KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)\)
  1. 6554292307
    \(t\)
  1. 4270680309
    \(\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}\)
 valid
12
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5781981178
    \(x^2 - y^2 = (x+y)(x-y)\)
 no validation is available for declarations
13
  • 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}.
  1. 5781981178
    \(x^2 - y^2 = (x+y)(x-y)\)
  1. 4319470443
    \(v_2\)
  1. 8173074178
    \(x\)
  1. 5239755033
    \(v_1\)
  1. 1025759423
    \(y\)
  1. 4648451961
    \(v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)\)
 LHS diff is -pdg0001452**2 + pdg0001464**2 + pdg0002473**2 - pdg0004770**2 RHS diff is -(pdg0001452 - pdg0001464)*(pdg0001452 + pdg0001464) + (pdg0002473 - pdg0004770)*(pdg0002473 + pdg0004770)
14
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4270680309
    \(\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}\)
  1. 4648451961
    \(v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)\)
  1. 9356924046
    \(\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}\)
 LHS diff is (-pdg0001352 + pdg0001467*(-pdg0002473**2 + pdg0004770**2) + pdg0001955)/pdg0001467 RHS diff is (-2*pdg0001467 + pdg0005156)*(pdg0002473 - pdg0004770)*(pdg0002473 + pdg0004770)/(2*pdg0001467)
15
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 9397152918
    \(v = \frac{v_1 + v_2}{2}\)
 no validation is available for declarations
16
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2857430695
    \(a = \frac{v_2 - v_1}{t}\)
 no validation is available for declarations
17
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 9356924046
    \(\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}\)
  1. 9397152918
    \(v = \frac{v_1 + v_2}{2}\)
  1. 7735737409
    \(\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}\)
 LHS diff is (-pdg0001352 + pdg0001357*pdg0001467 + pdg0001955)/pdg0001467 RHS diff is (pdg0001357*pdg0005156*(pdg0002473 - pdg0004770) + pdg0001467*(pdg0002473 + pdg0004770)/2)/pdg0001467
18
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 7735737409
    \(\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}\)
  1. 2857430695
    \(a = \frac{v_2 - v_1}{t}\)
  1. 4784793837
    \(\frac{KE_2 - KE_1}{t} = m v a\)
 LHS diff is (-pdg0001352 + pdg0001467*pdg0009140 + pdg0001955)/pdg0001467 RHS diff is (-pdg0001357*pdg0001467*pdg0005156*pdg0009140 - pdg0002473 + pdg0004770)/pdg0001467
19
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5345738321
    \(F = m a\)
 no validation is available for declarations
20
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4784793837
    \(\frac{KE_2 - KE_1}{t} = m v a\)
  1. 5345738321
    \(F = m a\)
  1. 2186083170
    \(\frac{KE_2 - KE_1}{t} = v F\)
 LHS diff is (-pdg0001352 + pdg0001467*pdg0004202 + pdg0001955)/pdg0001467 RHS diff is -pdg0001357*pdg0004202 + pdg0005156*pdg0009140
21
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 6715248283
    \(PE = -F x\)
 no validation is available for declarations
22
  • 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}.
  1. 6715248283
    \(PE = -F x\)
  1. 6941869627
    \(PE\)
  1. 4755369593
    \(x_2\)
  1. 4188639044
    \(x\)
  1. 4522137851
    \(PE_2\)
  1. 2431507955
    \(PE_2 = -F x_2\)
 LHS diff is pdg0004930 - pdg0008849 RHS diff is pdg0004202*(pdg0005467 - pdg0008849)
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}.
  1. 6715248283
    \(PE = -F x\)
  1. 1010267783
    \(PE\)
  1. 1552869972
    \(x_1\)
  1. 4218009993
    \(x\)
  1. 6749533119
    \(PE_1\)
  1. 4669290568
    \(PE_1 = -F x_1\)
 LHS diff is -pdg0004093 + pdg0004930 RHS diff is pdg0004202*(pdg0003852 - pdg0004093)
24
  • 0000111222: subtract expr 1 from expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Subtract Eq.~\ref{eq:#1} from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.
  1. 2431507955
    \(PE_2 = -F x_2\)
  1. 4669290568
    \(PE_1 = -F x_1\)
  1. 7734996511
    \(PE_2 - PE_1 = -F ( x_2 - x_1 )\)
 LHS diff is 2*pdg0004093 - 2*pdg0008849 RHS diff is 2*pdg0004202*(-pdg0003852 + pdg0005467)
25
  • 0000111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 7734996511
    \(PE_2 - PE_1 = -F ( x_2 - x_1 )\)
  1. 2016063530
    \(t\)
  1. 7267155233
    \(\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)\)
 valid
26
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 9337785146
    \(v = \frac{x_2 - x_1}{t}\)
 no validation is available for declarations
27
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 7267155233
    \(\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)\)
  1. 9337785146
    \(v = \frac{x_2 - x_1}{t}\)
  1. 4872970974
    \(\frac{PE_2 - PE_1}{t} = -F v\)
 LHS diff is (pdg0001357*pdg0001467 + pdg0004093 - pdg0008849)/pdg0001467 RHS diff is (pdg0001357*pdg0001467*pdg0004202 - pdg0003852 + pdg0005467)/pdg0001467
28
  • 0000111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 5514556106
    \(E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)\)
  1. 2081689540
    \(t\)
  1. 2770069250
    \(\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}\)
 valid
29
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 2770069250
    \(\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}\)
  1. 4872970974
    \(\frac{PE_2 - PE_1}{t} = -F v\)
  1. 3591237106
    \(\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v\)
 LHS diff is (-pdg0004093 - pdg0004550 + pdg0008849 + pg5579)/pdg0001467 RHS diff is (-pdg0001352 + pdg0001955)/pdg0001467
30
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 3591237106
    \(\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v\)
  1. 2186083170
    \(\frac{KE_2 - KE_1}{t} = v F\)
  1. 1772416655
    \(\frac{E_2 - E_1}{t} = v F - F v\)
 LHS diff is (pdg0001352 - pdg0001955 - pdg0004550 + pdg0005579)/pdg0001467 RHS diff is pdg0001357*pdg0004202
31
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1772416655
    \(\frac{E_2 - E_1}{t} = v F - F v\)
  1. 1809909100
    \(\frac{E_2 - E_1}{t} = 0\)
 valid
32
  • 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. 1809909100
    \(\frac{E_2 - E_1}{t} = 0\)
  1. 5778176146
    \(t\)
  1. 3806977900
    \(E_2 - E_1 = 0\)
 valid
33
  • 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}.
  1. 3806977900
    \(E_2 - E_1 = 0\)
  1. 5960438249
    \(E_1\)
  1. 8558338742
    \(E_2 = E_1\)
 valid
34
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 8558338742
    \(E_2 = E_1\)
 no validation is available for declarations

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