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review derivation: work and force and energy

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
8 substitute LHS of two expressions into expr
  1. 7676652285; locally 8207477:
    \(KE_2 = \frac{1}{2} m v_2^2\)
    \(pdg_{1352} = \frac{pdg_{4770}^{2} pdg_{5156}}{2}\)
  2. 4928007622; locally 8883350:
    \(KE_1 = \frac{1}{2} m v_1^2\)
    \(pdg_{1955} = \frac{pdg_{2473}^{2} pdg_{5156}}{2}\)
  3. 4811121942; locally 4236963:
    \(W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2\)
    \(pdg_{6789} = - \frac{pdg_{2473}^{2} pdg_{5156}}{2} + \frac{pdg_{4770}^{2} pdg_{5156}}{2}\)
  1. 3360172339; locally 4943050:
    \(W = KE_2 - KE_1\)
    \(pdg_{6789} = pdg_{1352} - pdg_{1955}\)
failed 7676652285:
4928007622:
4811121942:
3360172339:
7676652285:
4928007622:
4811121942:
3360172339:
2 integrate
  1. 1590774089; locally 2237799:
    \(dW = F dx\)
    \(pdg_{9398} = pdg_{4202} pdg_{9199}\)
  1. 5542528160; locally 2565189:
    \(\int dW = F \int_0^x dx\)
    \(\int 1\, dpdg_{6789} = pdg_{4202} \int\limits_{0}^{pdg_{4037}} 1\, dpdg_{4037}\)
no check performed 1590774089:
5542528160:
1590774089:
5542528160:
7 simplify
  1. 9413699705; locally 6760874:
    \(W = m a \frac{v_2^2 - v_1^2}{2 a}\)
    \(pdg_{6789} = pdg_{5156} \left(- \frac{pdg_{2473}^{2}}{2} + \frac{pdg_{4770}^{2}}{2}\right)\)
  1. 4811121942; locally 4236963:
    \(W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2\)
    \(pdg_{6789} = - \frac{pdg_{2473}^{2} pdg_{5156}}{2} + \frac{pdg_{4770}^{2} pdg_{5156}}{2}\)
valid 9413699705: dimensions are consistent
4811121942:
9413699705: N/A
4811121942:
9 declare final expr
  1. 3360172339; locally 4943050:
    \(W = KE_2 - KE_1\)
    \(pdg_{6789} = pdg_{1352} - pdg_{1955}\)
no validation is available for declarations 3360172339:
3360172339:
3 evaluate definite integral
  1. 5542528160; locally 2565189:
    \(\int dW = F \int_0^x dx\)
    \(\int 1\, dpdg_{6789} = pdg_{4202} \int\limits_{0}^{pdg_{4037}} 1\, dpdg_{4037}\)
  1. 3512166162; locally 7362045:
    \(W = F x\)
    \(pdg_{6789} = pdg_{4037} pdg_{4202}\)
valid 5542528160:
3512166162:
5542528160:
3512166162:
1 declare initial expr
  1. 1590774089; locally 2237799:
    \(dW = F dx\)
    \(pdg_{9398} = pdg_{4202} pdg_{9199}\)
no validation is available for declarations 1590774089:
1590774089:
4 substitute LHS of expr 1 into expr 2
  1. 3512166162; locally 7362045:
    \(W = F x\)
    \(pdg_{6789} = pdg_{4037} pdg_{4202}\)
  2. 5345738321; locally 3086821:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
  1. 8953094349; locally 6167182:
    \(W = m a x\)
    \(pdg_{6789} = pdg_{4037} pdg_{5156} pdg_{9140}\)
LHS diff is pdg4202 - pdg6789 RHS diff is pdg5156*pdg9140*(1 - pdg4037) 3512166162:
5345738321:
8953094349: dimensions are consistent
 3512166162:
5345738321:
8953094349: N/A
5 change three variables in expr
  1. 5611024898; locally 4741344:
    \(d = \frac{1}{2 a} (v^2 - v_0^2)\)
    \(pdg_{1943} = \frac{pdg_{1357}^{2} - pdg_{5153}^{2}}{2 pdg_{9140}}\)
  1. 9623791270:
    \(d\)
    \(pdg_{4037}\)
  2. 8111389082:
    \(x\)
    \(pdg_{4037}\)
  3. 3652511721:
    \(v\)
    \(pdg_{1357}\)
  4. 6701855578:
    \(v_2\)
    \(pdg_{4770}\)
  5. 5398681502:
    \(v\)
    \(pdg_{1357}\)
  6. 3183197515:
    \(v_1\)
    \(pdg_{2473}\)
  1. 3253234559; locally 5997798:
    \(x = \frac{v_2^2 - v_1^2}{2 a}\)
    \(pdg_{4037} = \frac{- pdg_{2473}^{2} + pdg_{4770}^{2}}{2 pdg_{9140}}\)
LHS diff is pdg1943 - pdg4037 RHS diff is (pdg2473**2 - pdg5153**2)/(2*pdg9140) 5611024898:
3253234559: dimensions are consistent
 5611024898:
3253234559: N/A
6 substitute LHS of expr 1 into expr 2
  1. 3253234559; locally 5997798:
    \(x = \frac{v_2^2 - v_1^2}{2 a}\)
    \(pdg_{4037} = \frac{- pdg_{2473}^{2} + pdg_{4770}^{2}}{2 pdg_{9140}}\)
  2. 8953094349; locally 6167182:
    \(W = m a x\)
    \(pdg_{6789} = pdg_{4037} pdg_{5156} pdg_{9140}\)
  1. 9413699705; locally 6760874:
    \(W = m a \frac{v_2^2 - v_1^2}{2 a}\)
    \(pdg_{6789} = pdg_{5156} \left(- \frac{pdg_{2473}^{2}}{2} + \frac{pdg_{4770}^{2}}{2}\right)\)
valid 3253234559: dimensions are consistent
8953094349: dimensions are consistent
9413699705: dimensions are consistent
 3253234559: N/A
8953094349: N/A
9413699705: N/A
Physics Derivation Graph: Steps for work and force and energy

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
1943 variable d
\(d\)		 ['real']
  • length: 1
 displacement 25
9199 variable dx
\(dx\)		 ['real']
  • length: 1
 15
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
1955 variable KE_1
\(KE_1\)		 real
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 17
2473 variable v_1
\(v_1\)		 real
  • length: 1
  • time: -1
 velocity 1 14
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
1352 variable KE_2
\(KE_2\)		 real
  • length: 2
  • mass: 1
  • time: -2
 kinetic energy 14
6789 variable W
\(W\)		 real
  • length: 2
  • mass: 1
  • time: -2
 work 10
5153 variable v_0
\(v_0\)		 ['real']
  • length: 1
  • time: -1
 initial velocity 44
4770 variable v_2
\(v_2\)		 real
  • length: 1
  • time: -1
 velocity 2 14
4202 variable F
\(F\)		 ['real']
  • length: 1
  • mass: 1
  • time: -2
 force 21
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
9398 variable dW
\(dW\)		 real
  • length: 2
  • mass: 1
  • time: -2
 differential work
  • str_note
2
9140 variable a
\(a\)		 ['real']
  • length: 1
  • time: -2
 acceleration 31
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