## review derivation: time invariant force conserves energy

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
27 substitute RHS of expr 1 into expr 2
1. 9337785146; locally 6154610:
$$v = \frac{x_2 - x_1}{t}$$
$$pdg_{1357} = \frac{- pdg_{3852} + pdg_{5467}}{pdg_{1467}}$$
2. 7267155233; locally 7539016:
$$\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - \frac{pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)}{pdg_{1467}}$$
1. 4872970974; locally 9383749:
$$\frac{PE_2 - PE_1}{t} = -F v$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - pdg_{1357} pdg_{4202}$$
valid 9337785146:
7267155233:
4872970974:
9337785146:
7267155233:
4872970974:
14 substitute RHS of expr 1 into expr 2
1. 4648451961; locally 8696678:
$$v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)$$
$$- pdg_{2473}^{2} + pdg_{4770}^{2} = \left(- pdg_{2473} + pdg_{4770}\right) \left(pdg_{2473} + pdg_{4770}\right)$$
2. 4270680309; locally 3040361:
$$\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2 pdg_{1467}}$$
1. 9356924046; locally 6246951:
$$\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right) \left(\frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}\right)}{pdg_{1467}}$$
valid 4648451961:
4270680309:
9356924046:
4648451961:
4270680309:
9356924046:
18 substitute RHS of expr 1 into expr 2
1. 2857430695; locally 6973462:
$$a = \frac{v_2 - v_1}{t}$$
$$pdg_{9140} = \frac{- pdg_{2473} + pdg_{4770}}{pdg_{1467}}$$
2. 7735737409; locally 6733685:
$$\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{1357} pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right)}{pdg_{1467}}$$
1. 4784793837; locally 4876963:
$$\frac{KE_2 - KE_1}{t} = m v a$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{5156} pdg_{9140}$$
valid 2857430695:
7735737409:
4784793837:
2857430695:
7735737409:
4784793837:
31 simplify
1. 1772416655; locally 5300304:
$$\frac{E_2 - E_1}{t} = v F - F v$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
1. 1809909100; locally 6495233:
$$\frac{E_2 - E_1}{t} = 0$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
valid 1772416655: error for dim with 1772416655
1809909100:
1772416655: N/A
1809909100:
32 multiply both sides by
1. 1809909100; locally 6495233:
$$\frac{E_2 - E_1}{t} = 0$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
1. 5778176146:
$$t$$
$$pdg_{1467}$$
1. 3806977900; locally 2075807:
$$E_2 - E_1 = 0$$
$$pdg_{4550} - pdg_{5579} = 0$$
valid 1809909100:
3806977900:
1809909100:
3806977900:
6 declare initial expr
1. 8357234146; locally 6559987:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
no validation is available for declarations 8357234146:
8357234146:
17 substitute RHS of expr 1 into expr 2
1. 9397152918; locally 3484339:
$$v = \frac{v_1 + v_2}{2}$$
$$pdg_{1357} = \frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}$$
2. 9356924046; locally 6246951:
$$\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right) \left(\frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}\right)}{pdg_{1467}}$$
1. 7735737409; locally 6733685:
$$\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{1357} pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right)}{pdg_{1467}}$$
valid 9397152918:
9356924046:
7735737409:
9397152918:
9356924046:
7735737409:
29 substitute RHS of expr 1 into expr 2
1. 4872970974; locally 9383749:
$$\frac{PE_2 - PE_1}{t} = -F v$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - pdg_{1357} pdg_{4202}$$
2. 2770069250; locally 2692856:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} + \frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}}$$
1. 3591237106; locally 9714818:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v$$
$$\frac{pdg_{4550} - pg_{5579}}{pdg_{1467}} = - pdg_{1357} pdg_{4202} + \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}}$$
LHS diff is (-pdg5579 + pg5579)/pdg1467 RHS diff is (pdg1357*pdg1467*pdg4202 - pdg4093 + pdg8849)/pdg1467 4872970974:
2770069250:
3591237106: error for dim with 3591237106
4872970974:
2770069250:
3591237106: N/A
7 change two variables in expr
1. 8357234146; locally 6559987:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 6383056612:
$$KE$$
$$pdg_{4929}$$
2. 6838659900:
$$KE_2$$
$$pdg_{1352}$$
3. 9305761407:
$$v$$
$$pdg_{1357}$$
4. 5011888122:
$$v_2$$
$$pdg_{4770}$$
1. 7676652285; locally 6632540:
$$KE_2 = \frac{1}{2} m v_2^2$$
$$pdg_{1352} = \frac{pdg_{4770}^{2} pdg_{5156}}{2}$$
valid 8357234146:
7676652285:
8357234146:
7676652285:
11 divide both sides by
1. 5733146966; locally 9602854:
$$KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)$$
$$pdg_{1352} - pdg_{1955} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2}$$
1. 6554292307:
$$t$$
$$pdg_{1467}$$
1. 4270680309; locally 3040361:
$$\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2 pdg_{1467}}$$
valid 5733146966:
4270680309:
5733146966:
4270680309:
1 declare initial expr
1. 5136652623; locally 8844119:
$$E = KE + PE$$
$$pdg_{4931} = pdg_{4929} + pdg_{4930}$$
no validation is available for declarations 5136652623:
5136652623:
2 change three variables in expr
1. 5136652623; locally 8844119:
$$E = KE + PE$$
$$pdg_{4931} = pdg_{4929} + pdg_{4930}$$
1. 1258245373:
$$E$$
$$pdg_{4931}$$
2. 2344320475:
$$E_2$$
$$pdg_{4550}$$
3. 6383056612:
$$KE$$
$$pdg_{4929}$$
4. 7939947931:
$$KE_2$$
$$pdg_{1352}$$
5. 5075406409:
$$PE$$
$$pdg_{4930}$$
6. 5803210729:
$$PE_2$$
$$pdg_{8849}$$
1. 7875206161; locally 5642407:
$$E_2 = KE_2 + PE_2$$
$$pdg_{4550} = pdg_{1352} + pdg_{8849}$$
valid 5136652623:
7875206161:
5136652623:
7875206161:
28 divide both sides by
1. 5514556106; locally 2443387:
$$E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)$$
$$pdg_{4550} - pdg_{5579} = pdg_{1352} - pdg_{1955} - pdg_{4093} + pdg_{8849}$$
1. 2081689540:
$$t$$
$$pdg_{1467}$$
1. 2770069250; locally 2692856:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} + \frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}}$$
valid 5514556106:
2770069250:
5514556106:
2770069250:
10 subtract expr 1 from expr 2
1. 4928007622; locally 4208138:
$$KE_1 = \frac{1}{2} m v_1^2$$
$$pdg_{1955} = \frac{pdg_{2473}^{2} pdg_{5156}}{2}$$
2. 7676652285; locally 6632540:
$$KE_2 = \frac{1}{2} m v_2^2$$
$$pdg_{1352} = \frac{pdg_{4770}^{2} pdg_{5156}}{2}$$
1. 5733146966; locally 9602854:
$$KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)$$
$$pdg_{1352} - pdg_{1955} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2}$$
valid 4928007622:
7676652285:
5733146966:
4928007622:
7676652285:
5733146966:
16 declare initial expr
1. 2857430695; locally 6973462:
$$a = \frac{v_2 - v_1}{t}$$
$$pdg_{9140} = \frac{- pdg_{2473} + pdg_{4770}}{pdg_{1467}}$$
no validation is available for declarations 2857430695:
2857430695:
25 divide both sides by
1. 7734996511; locally 1550851:
$$PE_2 - PE_1 = -F ( x_2 - x_1 )$$
$$- pdg_{4093} + pdg_{8849} = - pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)$$
1. 2016063530:
$$t$$
$$pdg_{1467}$$
1. 7267155233; locally 7539016:
$$\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - \frac{pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)}{pdg_{1467}}$$
valid 7734996511:
7267155233:
7734996511:
7267155233:
23 change two variables in expr
1. 6715248283; locally 8497204:
$$PE = -F x$$
$$pdg_{4930} = - pdg_{4037} pdg_{4202}$$
1. 3809726424:
$$PE$$
$$pdg_{4930}$$
2. 6749533119:
$$PE_1$$
$$pdg_{4093}$$
3. 4218009993:
$$x$$
$$pdg_{4037}$$
4. 1552869972:
$$x_1$$
$$pdg_{3852}$$
1. 4669290568; locally 9081932:
$$PE_1 = -F x_1$$
$$pdg_{4093} = - pdg_{3852} pdg_{4202}$$
valid 6715248283:
4669290568:
6715248283:
4669290568:
19 declare initial expr
1. 5345738321; locally 8447573:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
no validation is available for declarations 5345738321:
5345738321:
22 change two variables in expr
1. 6715248283; locally 8497204:
$$PE = -F x$$
$$pdg_{4930} = - pdg_{4037} pdg_{4202}$$
1. 5075406409:
$$PE$$
$$pdg_{4930}$$
2. 4522137851:
$$PE_2$$
$$pdg_{8849}$$
3. 4188639044:
$$x$$
$$pdg_{4037}$$
4. 4755369593:
$$x_2$$
$$pdg_{5467}$$
1. 2431507955; locally 3988671:
$$PE_2 = -F x_2$$
$$pdg_{8849} = - pdg_{4202} pdg_{5467}$$
valid 6715248283:
2431507955:
6715248283:
2431507955:
assumes constant force
5 subtract expr 1 from expr 2
1. 4303372136; locally 1298003:
$$E_1 = KE_1 + PE_1$$
$$pdg_{5579} = pdg_{1955} + pdg_{4093}$$
2. 7875206161; locally 5642407:
$$E_2 = KE_2 + PE_2$$
$$pdg_{4550} = pdg_{1352} + pdg_{8849}$$
1. 5514556106; locally 2443387:
$$E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)$$
$$pdg_{4550} - pdg_{5579} = pdg_{1352} - pdg_{1955} - pdg_{4093} + pdg_{8849}$$
valid 4303372136:
7875206161:
5514556106:
4303372136:
7875206161:
5514556106:
34 declare final expr
1. 8558338742; locally 1781127:
$$E_2 = E_1$$
$$pdg_{4550} = pdg_{5579}$$
no validation is available for declarations 8558338742:
8558338742:
20 substitute RHS of expr 1 into expr 2
1. 5345738321; locally 8447573:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
2. 4784793837; locally 4876963:
$$\frac{KE_2 - KE_1}{t} = m v a$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{5156} pdg_{9140}$$
1. 2186083170; locally 7034924:
$$\frac{KE_2 - KE_1}{t} = v F$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{4202}$$
valid 5345738321:
4784793837:
2186083170: dimensions are consistent
5345738321:
4784793837:
2186083170: N/A
12 declare initial expr
1. 5781981178; locally 2776565:
$$x^2 - y^2 = (x+y)(x-y)$$
$$- pdg_{1452}^{2} + pdg_{1464}^{2} = \left(- pdg_{1452} + pdg_{1464}\right) \left(pdg_{1452} + pdg_{1464}\right)$$
no validation is available for declarations 5781981178:
5781981178:
8 change two variables in expr
1. 8357234146; locally 6559987:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 4147101187:
$$KE$$
$$pdg_{4929}$$
2. 6964468708:
$$KE_1$$
$$pdg_{1955}$$
3. 5398681503:
$$v$$
$$pdg_{1357}$$
4. 3105350101:
$$v_1$$
$$pdg_{2473}$$
1. 4928007622; locally 4208138:
$$KE_1 = \frac{1}{2} m v_1^2$$
$$pdg_{1955} = \frac{pdg_{2473}^{2} pdg_{5156}}{2}$$
valid 8357234146:
4928007622:
8357234146:
4928007622:
21 declare initial expr
1. 6715248283; locally 8497204:
$$PE = -F x$$
$$pdg_{4930} = - pdg_{4037} pdg_{4202}$$
no validation is available for declarations 6715248283:
6715248283:
24 subtract expr 1 from expr 2
1. 4669290568; locally 9081932:
$$PE_1 = -F x_1$$
$$pdg_{4093} = - pdg_{3852} pdg_{4202}$$
2. 2431507955; locally 3988671:
$$PE_2 = -F x_2$$
$$pdg_{8849} = - pdg_{4202} pdg_{5467}$$
1. 7734996511; locally 1550851:
$$PE_2 - PE_1 = -F ( x_2 - x_1 )$$
$$- pdg_{4093} + pdg_{8849} = - pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)$$
valid 4669290568:
2431507955:
7734996511:
4669290568:
2431507955:
7734996511:
33 add X to both sides
1. 3806977900; locally 2075807:
$$E_2 - E_1 = 0$$
$$pdg_{4550} - pdg_{5579} = 0$$
1. 5960438249:
$$E_1$$
$$pdg_{5579}$$
1. 8558338742; locally 1781127:
$$E_2 = E_1$$
$$pdg_{4550} = pdg_{5579}$$
valid 3806977900:
8558338742:
3806977900:
8558338742:
13 change two variables in expr
1. 5781981178; locally 2776565:
$$x^2 - y^2 = (x+y)(x-y)$$
$$- pdg_{1452}^{2} + pdg_{1464}^{2} = \left(- pdg_{1452} + pdg_{1464}\right) \left(pdg_{1452} + pdg_{1464}\right)$$
1. 1025759423:
$$y$$
$$pdg_{1452}$$
2. 5239755033:
$$v_1$$
$$pdg_{2473}$$
3. 8173074178:
$$x$$
$$pdg_{1464}$$
4. 4319470443:
$$v_2$$
$$pdg_{4770}$$
1. 4648451961; locally 8696678:
$$v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)$$
$$- pdg_{2473}^{2} + pdg_{4770}^{2} = \left(- pdg_{2473} + pdg_{4770}\right) \left(pdg_{2473} + pdg_{4770}\right)$$
valid 5781981178:
4648451961:
5781981178:
4648451961:
26 declare initial expr
1. 9337785146; locally 6154610:
$$v = \frac{x_2 - x_1}{t}$$
$$pdg_{1357} = \frac{- pdg_{3852} + pdg_{5467}}{pdg_{1467}}$$
no validation is available for declarations 9337785146:
9337785146:
15 declare initial expr
1. 9397152918; locally 3484339:
$$v = \frac{v_1 + v_2}{2}$$
$$pdg_{1357} = \frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}$$
no validation is available for declarations 9397152918:
9397152918:
3 change three variables in expr
1. 5136652623; locally 8844119:
$$E = KE + PE$$
$$pdg_{4931} = pdg_{4929} + pdg_{4930}$$
1. 3749492596:
$$E$$
$$pdg_{4931}$$
2. 4213426349:
$$E_1$$
$$pdg_{5579}$$
3. 4147101187:
$$KE$$
$$pdg_{4929}$$
4. 1092872200:
$$KE_1$$
$$pdg_{1955}$$
5. 3809726424:
$$PE$$
$$pdg_{4930}$$
6. 8061701434:
$$PE_1$$
$$pdg_{4093}$$
1. 4303372136; locally 1298003:
$$E_1 = KE_1 + PE_1$$
$$pdg_{5579} = pdg_{1955} + pdg_{4093}$$
valid 5136652623:
4303372136:
5136652623:
4303372136:
30 substitute RHS of expr 1 into expr 2
1. 2186083170; locally 7034924:
$$\frac{KE_2 - KE_1}{t} = v F$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{4202}$$
2. 3591237106; locally 9714818:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v$$
$$\frac{pdg_{4550} - pg_{5579}}{pdg_{1467}} = - pdg_{1357} pdg_{4202} + \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}}$$
1. 1772416655; locally 5300304:
$$\frac{E_2 - E_1}{t} = v F - F v$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
LHS diff is (pdg5579 - pg5579)/pdg1467 RHS diff is 0 2186083170: dimensions are consistent
3591237106: error for dim with 3591237106
1772416655: error for dim with 1772416655
2186083170: N/A
3591237106: N/A
1772416655: N/A
Physics Derivation Graph: Steps for time invariant force conserves energy

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
4037 variable x
$$x$$
['real']
• length: 1
position
53
3852 variable x_1
$$x_1$$
real
• length: 1
position 1
• str_note
5
5467 variable x_2
$$x_2$$
real
• length: 1
position 2
• str_note
5
4929 variable KE
$$KE$$
['real']
• length: 2
• mass: 1
• time: -2
kinetic energy
7
4550 variable E_2
$$E_2$$
real
• length: 2
• mass: 1
• time: -2
energy 2
• str_note
9
5579 variable E_1
$$E_1$$
real
• length: 2
• mass: 1
• time: -2
energy 1
• str_note
9
4770 variable v_2
$$v_2$$
real
• length: 1
• time: -1
velocity 2
14
1452 variable y
$$y$$
['real'] dimensionless 13
1464 variable x
$$x$$
['real'] dimensionless 140
8849 variable PE_2
$$PE_2$$
real
• length: 2
• mass: 1
• time: -2
potential energy
11
2473 variable v_1
$$v_1$$
real
• length: 1
• time: -1
velocity 1
14
4931 variable E
$$E$$
['real']
• length: 2
• mass: 1
• time: -2
energy
10
5156 variable m
$$m$$
['real']
• mass: 1
mass
69
1955 variable KE_1
$$KE_1$$
real
• length: 2
• mass: 1
• time: -2
kinetic energy
17
1357 variable v
$$v$$
['real']
• length: 1
• time: -1
velocity
83
4093 variable PE_1
$$PE_1$$
real
• length: 2
• mass: 1
• time: -2
kinetic energy
12
1467 variable t
$$t$$
['real']
• time: 1
time
121
1352 variable KE_2
$$KE_2$$
real
• length: 2
• mass: 1
• time: -2
kinetic energy
14
9140 variable a
$$a$$
['real']
• length: 1
• time: -2
acceleration 31
4930 variable PE
$$PE$$
['real']
• length: 2
• mass: 1
• time: -2
potential energy
4
4202 variable F
$$F$$
['real']
• length: 1
• mass: 1
• time: -2
force
21
MESSAGE:
• local variable 'all_df' referenced before assignment