Review Derivation: Physics Derivation Graph

Return to navigation page or list derivations

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 $

d3js visualization of steps and expressions in projectile path in 2D is parabolic

Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Actions: Edit Derivation

You must be logged in to edit the steps of derivation, edit the abstract, or rename the derivation.

Generate Tex file or PDF file

Delete Derivation and all associated steps

You must be logged in to delete the derivation.

timing of Neo4j queries:

0.057 seconds for pdg_app/to_review_derivation 0c34f5a1-7fdb-4007-bfab-df1e9163a9f5
0.014 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationffad6046-f0f3-4f6c-bdd3-10d94a646c8c
0.025 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDffad6046-f0f3-4f6c-bdd3-10d94a646c8c
0.047 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTffad6046-f0f3-4f6c-bdd3-10d94a646c8c
0.04 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDffad6046-f0f3-4f6c-bdd3-10d94a646c8c
0.021 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTffad6046-f0f3-4f6c-bdd3-10d94a646c8c
0.028 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepffad6046-f0f3-4f6c-bdd3-10d94a646c8c
0.196 seconds for pdg_app/ 0c34f5a1-7fdb-4007-bfab-df1e9163a9f5

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