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Review projectile path in 2D is parabolic

abstract: Using the 2D equations of motion, show that projectile path is second order polynomial of the form $y = a x^2 + b x + c$
reference:

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.			9882526611 $v_{0, x} t = x - x_0$	no validation is available for declarations
2	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}.	9882526611 $v_{0, x} t = x - x_0$	6050070428 $v_{0, x}$	3274926090 $t = \frac{x - x_0}{v_{0, x}}$	valid
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1405465835 $y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0$	no validation is available for declarations
4	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}.	1405465835 $y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0$ 3274926090 $t = \frac{x - x_0}{v_{0, x}}$		7354529102 $y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0$	LHS diff is pdg0001467 - pdg0005647 RHS diff is (-pdg0001469pdg00029582 + pdg00016492(pdg0001572 - pdg0004037)*2/2 + pdg0002958(-pdg0001572 + pdg0004037 + pdg0009431(pdg0001572 - pdg0004037)))/pdg0002958*2
Note about step 5: expression is a second order polynomial; projectile motion is parabolic
5	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	7354529102 $y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0$			no validation is available for declarations

Symbols used in projectile path in 2D is parabolic

0000001649: $ g $
0000001467: $ t $
0000009107: $ v_y $
0000002958: $ v_{0, x} $
0000009431: $ v_{0, y} $
0000004037: $ x $
0000001572: $ x_0 $
0000005647: $ y $
0000001469: $ y_0 $

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timing of Neo4j queries:

0.017 seconds for pdg_app/to_review_derivation e9915cf1-1e53-433f-98fd-382ca0211485
0.013 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation4203343d-c10b-484a-9e03-3feaf0677879
0.029 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_ID4203343d-c10b-484a-9e03-3feaf0677879
0.014 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUT4203343d-c10b-484a-9e03-3feaf0677879
0.033 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEED4203343d-c10b-484a-9e03-3feaf0677879
0.023 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUT4203343d-c10b-484a-9e03-3feaf0677879
0.022 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_step4203343d-c10b-484a-9e03-3feaf0677879
0.037 seconds for pdg_app/ e9915cf1-1e53-433f-98fd-382ca0211485

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.			9882526611 \(v_{0, x} t = x - x_0\)	no validation is available for declarations
2	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}.	9882526611 \(v_{0, x} t = x - x_0\)	6050070428 \(v_{0, x}\)	3274926090 \(t = \frac{x - x_0}{v_{0, x}}\)	valid
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1405465835 \(y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0\)	no validation is available for declarations
4	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}.	1405465835 \(y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0\) 3274926090 \(t = \frac{x - x_0}{v_{0, x}}\)		7354529102 \(y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0\)	LHS diff is pdg0001467 - pdg0005647 RHS diff is (-pdg0001469pdg00029582 + pdg00016492(pdg0001572 - pdg0004037)*2/2 + pdg0002958(-pdg0001572 + pdg0004037 + pdg0009431(pdg0001572 - pdg0004037)))/pdg0002958*2
Note about step 5: expression is a second order polynomial; projectile motion is parabolic
5	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	7354529102 \(y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0\)			no validation is available for declarations