Return to navigation page or list derivations

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
    \(t_f\)
  1. 2403773761
    \(t\)
  1. 8120663858
    \(y_f\)
  1. 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)
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
    \(y_f = 0\)
  1. 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
    \(y_0 = 0\)
  1. 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
    \(t_f\)
  1. 6732786762
    \(t\)
  1. 5194141542
    \(x_f\)
  1. 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
    \(x_f = v_0 t_f \cos(\theta) + x_0\)
  1. 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
    \(d = v_0 t_f \cos(\theta)\)
  1. 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
    \(x\)
  1. 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
    \(d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)\)
  1. 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
    \(d = \frac{v_0^2}{g} \sin(2 \theta)\)
  1. 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

Symbols used in angle of maximum distance for projectile motion

Steps and expressions for angle of maximum distance for projectile motion

d3js visualization of steps and expressions in angle of maximum distance for projectile motion


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Actions: Edit Derivation

Generate Tex file or PDF file
   xor   

Delete Derivation and all associated steps

This does not remove expressions, symbols, and operations.

timing of Neo4j queries: