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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:

0.006 seconds for pdg_app/to_review_derivation: node_properties, derivation 5375075
0.006 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation7136819
0.013 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID9404346
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0.008 seconds for compute/get_dict_of_steps_in_derivation: get_step_has_sequence_index7136819
0.006 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID3817400
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0.005 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID1444550
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0.007 seconds for pdg_app/to_review_derivation: get_list_of_step_dicts_in_this_derivation5375075
0.01 seconds for pdg_app/to_review_derivation: get_list_of_sequence_values_for_derivation_id5375075918264
0.041 seconds for pdg_app/to_review_derivation: get_nodes_of_type inference_rule5375075
0.012 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID8131447
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0.012 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID6117735
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0.013 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID9984624
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0.006 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID7054426
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0.01 seconds for compute/get_dict_of_steps_in_derivation: get_inference_rule_connected_to_step_ID5545214
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0.006 seconds for compute/get_dict_of_steps_in_derivation: get_expressions_from_step_id_and_expr_type, HAS_OUTPUT5545214

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