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Review angle of maximum distance for projectile motion

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. 9862900242
    \(y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)
 no validation is available for declarations
2
  • 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. 9862900242
    \(y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)
  1. 4162188238
    \(t_f\)
  2. 2403773761
    \(t\)
  3. 8120663858
    \(y_f\)
  4. 8406170337
    \(y\)
  1. 5379546684
    \(y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)
 LHS diff is pdg0005647 - pdg0007092 RHS diff is -pdg0001467**2*pdg0001649/2 + pdg0001467*pdg0005153*sin(pdg0001575) + pdg0001649*pdg0002467**2/2 - pdg0002467*pdg0005153*sin(pdg0001575)
Note about step 3: $y(t_f) = y_f = 0$
3
  • 0000111278: boundary condition
  • number of inputs: 1; feeds: 0; outputs: 1
  • Boundary condition: Eq.~\ref{eq:#2} when Eq.~\ref{eq#1}.
  1. 5373931751
    \(t = t_f\)
  1. 9112191201
    \(y_f = 0\)
 no validation is available for assumptions
4
  • 0000111355: LHS of expr 1 equals LHS of expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 9112191201
    \(y_f = 0\)
  2. 5379546684
    \(y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)
  1. 8198310977
    \(0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)
 valid
5
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 1650441634
    \(y_0 = 0\)
 no validation is available for declarations
6
  • 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. 1650441634
    \(y_0 = 0\)
  2. 8198310977
    \(0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)
  1. 1087417579
    \(0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)\)
 valid
7
  • 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. 1087417579
    \(0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)\)
  1. 4829590294
    \(t_f\)
  1. 2086924031
    \(0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)\)
 valid
8
  • 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. 2086924031
    \(0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)\)
  1. 6974054946
    \(\frac{1}{2} g t_f\)
  1. 1191796961
    \(\frac{1}{2} g t_f = v_0 \sin(\theta)\)
 valid
9
  • 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. 1191796961
    \(\frac{1}{2} g t_f = v_0 \sin(\theta)\)
  1. 2510804451
    \(2/g\)
  1. 4778077984
    \(t_f = \frac{2 v_0 \sin(\theta)}{g}\)
 valid
10
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5438722682
    \(x = v_0 t \cos(\theta) + x_0\)
 no validation is available for declarations
11
  • 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. 5438722682
    \(x = v_0 t \cos(\theta) + x_0\)
  1. 6463266449
    \(t_f\)
  2. 6732786762
    \(t\)
  3. 5194141542
    \(x_f\)
  4. 3273630811
    \(x\)
  1. 3485125659
    \(x_f = v_0 t_f \cos(\theta) + x_0\)
 LHS diff is -pdg0003652 + pdg0004037 RHS diff is pdg0005153*(pdg0001467 - pdg0002467)*cos(pdg0001575)
12
  • 0000111278: boundary condition
  • number of inputs: 1; feeds: 0; outputs: 1
  • Boundary condition: Eq.~\ref{eq:#2} when Eq.~\ref{eq#1}.
  1. 4370074654
    \(t = t_f\)
  1. 2378095808
    \(x_f = x_0 + d\)
 no validation is available for assumptions
13
  • 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. 3485125659
    \(x_f = v_0 t_f \cos(\theta) + x_0\)
  2. 2378095808
    \(x_f = x_0 + d\)
  1. 4268085801
    \(x_0 + d = v_0 t_f \cos(\theta) + x_0\)
 LHS diff is -pdg0001943 + pdg0002467*pdg0005153*cos(pdg0001575) RHS diff is pdg0001943 - pdg0002467*pdg0005153*cos(pdg0001575)
14
  • 0000111282: subtract X from both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4268085801
    \(x_0 + d = v_0 t_f \cos(\theta) + x_0\)
  1. 8072682558
    \(x_0\)
  1. 7233558441
    \(d = v_0 t_f \cos(\theta)\)
 valid
15
  • 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. 7233558441
    \(d = v_0 t_f \cos(\theta)\)
  2. 4778077984
    \(t_f = \frac{2 v_0 \sin(\theta)}{g}\)
  1. 2297105551
    \(d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)\)
 LHS diff is -pdg0001943 + pdg0002467 RHS diff is 2*pdg0005153*(-pdg0005153*cos(pdg0001575) + 1)*sin(pdg0001575)/pdg0001649
16
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2405307372
    \(\sin(2 x) = 2 \sin(x) \cos(x)\)
 no validation is available for declarations
17
  • 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. 2405307372
    \(\sin(2 x) = 2 \sin(x) \cos(x)\)
  1. 7214442790
    \(x\)
  2. 7587034465
    \(\theta\)
  1. 2519058903
    \(\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)\)
 valid
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. 2297105551
    \(d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)\)
  2. 2519058903
    \(\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)\)
  1. 8922441655
    \(d = \frac{v_0^2}{g} \sin(2 \theta)\)
 LHS diff is -pdg0001943 + sin(2*pdg0001575) RHS diff is (pdg0001649 - pdg0005153**2)*sin(2*pdg0001575)/pdg0001649
19
  • 0000111773: maximum of expression
  • number of inputs: 1; feeds: 1; outputs: 1
  • The maximum of Eq.~\ref{eq:#2} with respect to $#1$ is Eq.~\ref{eq:#3}
  1. 8922441655
    \(d = \frac{v_0^2}{g} \sin(2 \theta)\)
  1. 5667870149
    \(\theta\)
  1. 1541916015
    \(\theta = \frac{\pi}{4}\)
 recognized infrule but not yet supported
20
  • 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. 8922441655
    \(d = \frac{v_0^2}{g} \sin(2 \theta)\)
  2. 1541916015
    \(\theta = \frac{\pi}{4}\)
  1. 3607070319
    \(d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)\)
 LHS diff is pdg0001575 - pdg0001943 RHS diff is pdg0003141/4 - pdg0005153**2*sin(pdg0003141/2)/pdg0001649
21
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 3607070319
    \(d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)\)
  1. 5353282496
    \(d = \frac{v_0^2}{g}\)
 LHS diff is 0 RHS diff is pdg0005153**2*(sin(pdg0003141/2) - 1)/pdg0001649
22
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 5353282496
    \(d = \frac{v_0^2}{g}\)
no validation is available for declarations
23
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 1541916015
    \(\theta = \frac{\pi}{4}\)
no validation is available for declarations

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