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Review equations of motion in 2D (calculus)

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. 3169580383
    \(\vec{a} = \frac{d\vec{v}}{dt}\)
 no validation is available for declarations
3
  • 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. 8602512487
    \(\vec{a} = a_x \hat{x} + a_y \hat{y}\)
  1. 3169580383
    \(\vec{a} = \frac{d\vec{v}}{dt}\)
  1. 4158986868
    \(a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}\)
 Not evaluated due to missing term in SymPy
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. 4158986868
    \(a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}\)
  1. 5349866551
    \(\vec{v} = v_x \hat{x} + v_y \hat{y}\)
  1. 7729413831
    \(a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)\)
 Not evaluated due to missing term in SymPy
6
  • 0000111270: separate two vector components
  • number of inputs: 1; feeds: 0; outputs: 2
  • Separate two vector components in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}
  1. 7729413831
    \(a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)\)
  1. 8228733125
    \(a_y = \frac{d}{dt} v_y\)
  1. 1819663717
    \(a_x = \frac{d}{dt} v_x\)
 recognized infrule but not yet supported
7
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 9707028061
    \(a_x = 0\)
 no validation is available for declarations
8
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 2741489181
    \(a_y = -g\)
 no validation is available for declarations
9
  • 0000111791: assume N dimensions
  • number of inputs: 0; feeds: 1; outputs: 1
  • Assume $#1$ dimensions; decompose vector to be Eq.~\ref{eq:#2}.
  1. 3270039798
    \(2\)
  1. 8602512487
    \(\vec{a} = a_x \hat{x} + a_y \hat{y}\)
 no validation is available for assumptions
10
  • 0000111791: assume N dimensions
  • number of inputs: 0; feeds: 1; outputs: 1
  • Assume $#1$ dimensions; decompose vector to be Eq.~\ref{eq:#2}.
  1. 8880467139
    \(2\)
  1. 5349866551
    \(\vec{v} = v_x \hat{x} + v_y \hat{y}\)
 no validation is available for assumptions
11
  • 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. 1819663717
    \(a_x = \frac{d}{dt} v_x\)
  1. 9707028061
    \(a_x = 0\)
  1. 8750379055
    \(0 = \frac{d}{dt} v_x\)
 valid
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. 8228733125
    \(a_y = \frac{d}{dt} v_y\)
  1. 2741489181
    \(a_y = -g\)
  1. 1977955751
    \(-g = \frac{d}{dt} v_y\)
 LHS diff is pdg0001649 RHS diff is -pdg0001649
13
  • 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. 1977955751
    \(-g = \frac{d}{dt} v_y\)
  1. 6672141531
    \(dt\)
  1. 1702349646
    \(-g dt = d v_y\)
 LHS arithmetic error. Diff: pdg0001649*(dt - pdg0004711)
14
  • 0000111608: indefinite integration
  • number of inputs: 1; feeds: 0; outputs: 1
  • Indefinite integral of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1702349646
    \(-g dt = d v_y\)
  1. 8584698994
    \(-g \int dt = \int d v_y\)
 recognized infrule but not yet supported
15
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8584698994
    \(-g \int dt = \int d v_y\)
  1. 9973952056
    \(-g t = v_y - v_{0, y}\)
 LHS diff is -dt*g + pdg0001467*pdg0001649 RHS diff is pdg0005153 + pdg0005674 - pdg0009431
16
  • 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. 9973952056
    \(-g t = v_y - v_{0, y}\)
  1. 4167526462
    \(v_{0, y}\)
  1. 6572039835
    \(-g t + v_{0, y} = v_y\)
 RHS diff is -pdg0005153 - pdg0009107 + 2*pdg0009431
17
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 7252338326
    \(v_y = \frac{dy}{dt}\)
 no validation is available for declarations
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. 6572039835
    \(-g t + v_{0, y} = v_y\)
  1. 7252338326
    \(v_y = \frac{dy}{dt}\)
  1. 6204539227
    \(-g t + v_{0, y} = \frac{dy}{dt}\)
 LHS diff is pdg0001467*pdg0006277 + pdg0009107 - pdg0009431 RHS diff is 0
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}.
  1. 6204539227
    \(-g t + v_{0, y} = \frac{dy}{dt}\)
  1. 1614343171
    \(dt\)
  1. 8145337879
    \(-g t dt + v_{0, y} dt = dy\)
 LHS arithmetic error. Diff: pdg0001467*pdg0004711*(pdg0001649 - pdg0006277)
20
  • 0000111608: indefinite integration
  • number of inputs: 1; feeds: 0; outputs: 1
  • Indefinite integral of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8145337879
    \(-g t dt + v_{0, y} dt = dy\)
  1. 8808860551
    \(-g \int t dt + v_{0, y} \int dt = \int dy\)
 recognized infrule but not yet supported
21
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8808860551
    \(-g \int t dt + v_{0, y} \int dt = \int dy\)
  1. 2858549874
    \(- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0\)
 LHS diff is 0 RHS diff is pdg0001469
22
  • 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. 2858549874
    \(- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0\)
  1. 6098638221
    \(y_0\)
  1. 2461349007
    \(- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y\)
 valid
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. 8750379055
    \(0 = \frac{d}{dt} v_x\)
  1. 8717193282
    \(dt\)
  1. 1166310428
    \(0 dt = d v_x\)
 RHS arithmetic error. Diff: -pdg0005005
24
  • 0000111608: indefinite integration
  • number of inputs: 1; feeds: 0; outputs: 1
  • Indefinite integral of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1166310428
    \(0 dt = d v_x\)
  1. 2366691988
    \(\int 0 dt = \int d v_x\)
 recognized infrule but not yet supported
25
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 2366691988
    \(\int 0 dt = \int d v_x\)
  1. 1676472948
    \(0 = v_x - v_{0, x}\)
 LHS diff is 0 RHS diff is pdg0002958 + pdg0005005 - pdg0005505
26
  • 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. 1676472948
    \(0 = v_x - v_{0, x}\)
  1. 1439089569
    \(v_{0, x}\)
  1. 6134836751
    \(v_{0, x} = v_x\)
 valid
27
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8460820419
    \(v_x = \frac{dx}{dt}\)
 no validation is available for declarations
28
  • 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. 8460820419
    \(v_x = \frac{dx}{dt}\)
  1. 6134836751
    \(v_{0, x} = v_x\)
  1. 7455581657
    \(v_{0, x} = \frac{dx}{dt}\)
 valid
29
  • 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. 7455581657
    \(v_{0, x} = \frac{dx}{dt}\)
  1. 8607458157
    \(dt\)
  1. 1963253044
    \(v_{0, x} dt = dx\)
 RHS arithmetic error. Diff: -pdg0009199
30
  • 0000111608: indefinite integration
  • number of inputs: 1; feeds: 0; outputs: 1
  • Indefinite integral of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1963253044
    \(v_{0, x} dt = dx\)
  1. 3676159007
    \(v_{0, x} \int dt = \int dx\)
 recognized infrule but not yet supported
31
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 3676159007
    \(v_{0, x} \int dt = \int dx\)
  1. 9882526611
    \(v_{0, x} t = x - x_0\)
 LHS diff is 0 RHS diff is pdg0001464 + pdg0001572 - pdg0004037
32
  • 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. 9882526611
    \(v_{0, x} t = x - x_0\)
  1. 3182907803
    \(x_0\)
  1. 8486706976
    \(v_{0, x} t + x_0 = x\)
 valid
33
  • 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}.
  1. 8486706976
    \(v_{0, x} t + x_0 = x\)
  1. 1306360899
    \(x = v_{0, x} t + x_0\)
 valid
34
  • 0000111791: assume N dimensions
  • number of inputs: 0; feeds: 1; outputs: 1
  • Assume $#1$ dimensions; decompose vector to be Eq.~\ref{eq:#2}.
  1. 7049769409
    \(2\)
  1. 9341391925
    \(\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}\)
 no validation is available for assumptions
35
  • 0000111295: separate vector into two trigonometric ratios
  • number of inputs: 1; feeds: 1; outputs: 2
  • Separate vector in Eq.~\ref{eq:#2} into components related by angle $#1$; yields Eq.~\ref{eq:#3} and Eq.~\ref{eq:#4}.
  1. 9341391925
    \(\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}\)
  1. 6410818363
    \(\theta\)
  1. 7376526845
    \(\sin(\theta) = \frac{v_{0, y}}{v_0}\)
  1. 7391837535
    \(\cos(\theta) = \frac{v_{0, x}}{v_0}\)
 recognized infrule but not yet supported
36
  • 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. 7391837535
    \(\cos(\theta) = \frac{v_{0, x}}{v_0}\)
  1. 5868731041
    \(v_0\)
  1. 6083821265
    \(v_0 \cos(\theta) = v_{0, x}\)
 RHS arithmetic error. Diff: -pdg0002958 + pdg0005153**2/pdg0002958
37
  • 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. 1306360899
    \(x = v_{0, x} t + x_0\)
  1. 6083821265
    \(v_0 \cos(\theta) = v_{0, x}\)
  1. 5438722682
    \(x = v_0 t \cos(\theta) + x_0\)
 LHS diff is -pdg0004037 + pdg0005153*cos(pdg0001575) RHS diff is -pdg0001467*pdg0005153*cos(pdg0001575) - pdg0001572 + pdg0002958
38
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 5438722682
    \(x = v_0 t \cos(\theta) + x_0\)
 no validation is available for declarations
39
  • 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. 7376526845
    \(\sin(\theta) = \frac{v_{0, y}}{v_0}\)
  1. 5620558729
    \(v_0\)
  1. 8949329361
    \(v_0 \sin(\theta) = v_{0, y}\)
 RHS arithmetic error. Diff: pdg0005153**2/pdg0009431 - pdg0009431
40
  • 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}.
  1. 2461349007
    \(- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y\)
  1. 1405465835
    \(y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0\)
 LHS diff is pdg0001467*(-pdg0009107 + pdg0009431) RHS diff is 0
41
  • 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. 1405465835
    \(y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0\)
  1. 8949329361
    \(v_0 \sin(\theta) = v_{0, y}\)
  1. 9862900242
    \(y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)
 LHS diff is pdg0005153*sin(pdg0001575) - pdg0005647 RHS diff is pdg0001467**2*pdg0001649/2 - pdg0001467*pdg0005153*sin(pdg0001575) - pdg0001469 + pdg0009431
42
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 9862900242
    \(y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)
 no validation is available for declarations

Symbols used in equations of motion in 2D (calculus)

Steps and expressions for equations of motion in 2D (calculus)

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