## 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
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: dimensions are consistent
7354529102:
3274926090:
1405465835: N/A
7354529102:
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:
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:
7354529102:
expression is a second order polynomial; projecticle motion is parabolic
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:
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: dimensions are consistent
1405465835: N/A
Physics Derivation Graph: Steps for projectile path in 2D is parabolic

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
1572 variable x_0
$$x_0$$
['real']
• length: 1
initial position 11
9107 variable v_y
$$v_y$$
real
• length: 1
• time: -1
velocity along y axis
• str_note
7
1649 variable g
$$g$$
['real']
• length: 1
• time: -2
acceleration due to gravity
27
2958 variable v_{0, x}
$$v_{0, x}$$
['real']
• length: 1
• time: -1
initial velocity along x axis
15
4037 variable x
$$x$$
['real']
• length: 1
position
53
9431 variable v_{0, y}
$$v_{0, y}$$
['real']
• length: 1
• time: -1
initial velocity along y axis
12
1469 variable y_0
$$y_0$$
['real']
• length: 1
initial position 9
1467 variable t
$$t$$
['real']
• time: 1
time
121
5647 variable y
$$y$$
['real']
• length: 1
position
14
MESSAGE:
• local variable 'all_df' referenced before assignment