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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	111981: 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	111975: 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	111981: 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	111556: 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}.	3274926090 $t=\frac{x - x_0}{v_{0, x}}$ 1405465835 $y=- \frac{1}{2} g t^2 + v_{0, y} t + y_0$		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 0 RHS diff is (pdg0001572 - pdg0004037)(pdg0001649(pdg0001572 - pdg0004037)(pdg0001649 - 1) + 2pdg0002958(-pdg0009107 + pdg0009431))/(2pdg0002958**2)
5	111341: 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

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timing of Neo4j queries:

0.016 seconds for pdg_app/to_review_derivation: node_properties, derivation 2849803
0.009 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation8553288
0.013 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4864527
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4864527
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4864527
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4864527
0.007 seconds for compute/get_dict_of_steps_in_derivation: get_step_has_sequence_index8553288
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9376799
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9376799
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9376799
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9376799
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8932414
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT8932414
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED8932414
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT8932414
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9068567
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9068567
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9068567
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9068567
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3958625
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3958625
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3958625
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3958625
0.004 seconds for pdg_app/to_review_derivation: get_list_of_step_dicts_in_this_derivation2849803
0.007 seconds for pdg_app/to_review_derivation: get_list_of_sequence_values_for_derivation_id2849803918264
0.03 seconds for pdg_app/to_review_derivation: get_list_node_dicts_of_type inference_rule2849803
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2879105
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2879105
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2879105
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2879105
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule1651787
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT1651787
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED1651787
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT1651787
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7556197
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7556197
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7556197
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7556197
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5209972
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5209972
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5209972
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5209972
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5719850
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5719850
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5719850
0.029 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5719850
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4795558
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4795558
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4795558
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4795558
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9145869
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9145869
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9145869
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9145869
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6981954
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6981954
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6981954
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6981954
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8433749
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT8433749
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED8433749
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT8433749
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule1776794
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT1776794
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED1776794
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT1776794

step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: 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	111975: 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	111981: 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	111556: 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}.	3274926090 \(t=\frac{x - x_0}{v_{0, x}}\) 1405465835 \(y=- \frac{1}{2} g t^2 + v_{0, y} t + y_0\)		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 0 RHS diff is (pdg0001572 - pdg0004037)(pdg0001649(pdg0001572 - pdg0004037)(pdg0001649 - 1) + 2pdg0002958(-pdg0009107 + pdg0009431))/(2pdg0002958**2)
5	111341: 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