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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
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:
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:
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:
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:
3512166162:
5345738321:
8953094349:
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:
5611024898:
3253234559:
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:
8953094349:
9413699705:
3253234559:
8953094349:
9413699705:
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:
4811121942:
9413699705:
4811121942:
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: dimensions are consistent
7676652285:
4928007622:
4811121942:
3360172339: N/A
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: dimensions are consistent
3360172339: N/A
Physics Derivation Graph: Steps for work and force and energy

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
9398 variable dW
\(dW\)
real
  • length: 2
  • time: -2
  • mass: 1
differential work
  • str_note
2
6789 variable W
\(W\)
real
  • length: 2
  • time: -2
  • mass: 1
work 10
9140 variable a
\(a\)
['real']
  • length: 1
  • time: -2
acceleration 29
4770 variable v_2
\(v_2\)
real
  • length: 1
  • time: -1
velocity 2 14
1955 variable KE_1
\(KE_1\)
real
  • length: 2
  • time: -2
  • mass: 1
kinetic energy 17
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 80
4037 variable x
\(x\)
['real']
  • length: 1
position 47
2473 variable v_1
\(v_1\)
real
  • length: 1
  • time: -1
velocity 1 14
1943 variable d
\(d\)
['real']
  • length: 1
displacement 25
5153 variable v_0
\(v_0\)
['real']
  • length: 1
  • time: -1
initial velocity 44
1352 variable KE_2
\(KE_2\)
real
  • length: 2
  • time: -2
  • mass: 1
kinetic energy 14
4202 variable F
\(F\)
['real']
  • length: 1
  • time: -2
  • mass: 1
force 21
9199 variable dx
\(dx\)
['real']
  • length: 1
15
5156 variable m
\(m\)
['real']
  • mass: 1
mass 49
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