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review derivation: projectile path in 2D is parabolic

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Notes for this derivation:
Using the 2D equations of motion, show that projectile path is second order polynomial of the form y = a x^2 + b x + c

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 declare initial expr
  1. 9882526611; locally 4718749:
    \(v_{0, x} t = x - x_0\)
    \(pdg_{1467} pdg_{2958} = - pdg_{1572} + pdg_{4037}\)
no validation is available for declarations 9882526611:
9882526611:
2 divide both sides by
  1. 9882526611; locally 4718749:
    \(v_{0, x} t = x - x_0\)
    \(pdg_{1467} pdg_{2958} = - pdg_{1572} + pdg_{4037}\)
  1. 6050070428:
    \(v_{0, x}\)
    \(pdg_{2958}\)
  1. 3274926090; locally 8858248:
    \(t = \frac{x - x_0}{v_{0, x}}\)
    \(pdg_{1467} = \frac{- pdg_{1572} + pdg_{4037}}{pdg_{2958}}\)
valid 9882526611:
3274926090:
9882526611:
3274926090:
3 declare initial expr
  1. 1405465835; locally 5756391:
    \(y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0\)
    \(pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9107} + pdg_{1469}\)
no validation is available for declarations 1405465835:
1405465835:
4 substitute LHS of expr 1 into expr 2
  1. 3274926090; locally 8858248:
    \(t = \frac{x - x_0}{v_{0, x}}\)
    \(pdg_{1467} = \frac{- pdg_{1572} + pdg_{4037}}{pdg_{2958}}\)
  2. 1405465835; locally 5756391:
    \(y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0\)
    \(pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9107} + pdg_{1469}\)
  1. 7354529102; locally 9683207:
    \(y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0\)
    \(pdg_{5647} = pdg_{1469} - \frac{pdg_{1649}^{2} \left(- pdg_{1572} + pdg_{4037}\right)^{2}}{2 pdg_{2958}^{2}} + \frac{pdg_{9431} \left(- pdg_{1572} + pdg_{4037}\right)}{pdg_{2958}}\)
LHS diff is 0 RHS diff is (pdg1572 - pdg4037)*(pdg1649*(pdg1572 - pdg4037)*(pdg1649 - 1) + 2*pdg2958*(-pdg9107 + pdg9431))/(2*pdg2958**2) 3274926090:
1405465835:
7354529102: error for dim with 7354529102
3274926090:
1405465835:
7354529102: N/A
5 declare final expr
  1. 7354529102; locally 9683207:
    \(y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0\)
    \(pdg_{5647} = pdg_{1469} - \frac{pdg_{1649}^{2} \left(- pdg_{1572} + pdg_{4037}\right)^{2}}{2 pdg_{2958}^{2}} + \frac{pdg_{9431} \left(- pdg_{1572} + pdg_{4037}\right)}{pdg_{2958}}\)
no validation is available for declarations 7354529102: error for dim with 7354529102
7354529102: N/A
expression is a second order polynomial; projecticle motion is parabolic
Physics Derivation Graph: Steps for projectile path in 2D is parabolic

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1467 variable t
\(t\)
['real']
  • time: 1
time 115
4037 variable x
\(x\)
['real']
  • length: 1
position 47
1572 variable x_0
\(x_0\)
['real']
  • length: 1
initial position 11
5647 variable y
\(y\)
['real']
  • length: 1
position 13
2958 variable v_{0, x}
\(v_{0, x}\)
['real']
  • length: 1
  • time: -1
initial velocity along x axis 15
9431 variable v_{0, y}
\(v_{0, y}\)
['real']
  • length: 1
  • time: -1
initial velocity along y axis 12
1649 variable g
\(g\)
['real']
  • length: 1
  • time: -2
acceleration due to gravity 27
1469 variable y_0
\(y_0\)
['real']
  • length: 1
initial position 9
9107 variable v_y
\(v_y\)
real
  • length: 1
  • time: -1
velocity along y axis
  • str_note
7
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