## 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

symbol ID category latex scope dimension name value Used in derivations references
1352 variable KE_2
$$KE_2$$
real
• length: 2
• mass: 1
• time: -2
kinetic energy
14
4202 variable F
$$F$$
['real']
• length: 1
• mass: 1
• time: -2
force
21
9199 variable dx
$$dx$$
['real']
• length: 1
15
2473 variable v_1
$$v_1$$
real
• length: 1
• time: -1
velocity 1
14
4037 variable x
$$x$$
['real']
• length: 1
position
53
4770 variable v_2
$$v_2$$
real
• length: 1
• time: -1
velocity 2
14
1943 variable d
$$d$$
['real']
• length: 1
displacement
25
5156 variable m
$$m$$
['real']
• mass: 1
mass
57
1955 variable KE_1
$$KE_1$$
real
• length: 2
• mass: 1
• time: -2
kinetic energy
17
9140 variable a
$$a$$
['real']
• length: 1
• time: -2
acceleration 31
1357 variable v
$$v$$
['real']
• length: 1
• time: -1
velocity
80
9398 variable dW
$$dW$$
real
• length: 2
• mass: 1
• time: -2
differential work
• str_note
2
5153 variable v_0
$$v_0$$
['real']
• length: 1
• time: -1
initial velocity 44
6789 variable W
$$W$$
real
• length: 2
• mass: 1
• time: -2
work
10
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 1136729: 0