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review derivation: equations of motion in 1D with constant acceleration - SUVAT (algebra)

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Notes for this derivation:
https://en.wikipedia.org/wiki/Equations_of_motion

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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. 3366703541; locally 7864125:
    \(a = \frac{v - v_0}{t}\)
    \(pdg_{9140} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{1467}}\)
no validation is available for declarations 3366703541:
3366703541:
2 multiply both sides by
  1. 3366703541; locally 7864125:
    \(a = \frac{v - v_0}{t}\)
    \(pdg_{9140} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{1467}}\)
  1. 7083390553:
    \(t\)
    \(pdg_{1467}\)
  1. 4748157455; locally 5666935:
    \(a t = v - v_0\)
    \(pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}\)
valid 3366703541:
4748157455:
3366703541:
4748157455:
3 add X to both sides
  1. 4748157455; locally 5666935:
    \(a t = v - v_0\)
    \(pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}\)
  1. 6417359412:
    \(v_0\)
    \(pdg_{5153}\)
  1. 4798787814; locally 3386860:
    \(a t + v_0 = v\)
    \(pdg_{1467} pdg_{9140} + pdg_{5153} = pdg_{1357}\)
valid 4748157455:
4798787814:
4748157455:
4798787814:
4 swap LHS with RHS
  1. 4798787814; locally 3386860:
    \(a t + v_0 = v\)
    \(pdg_{1467} pdg_{9140} + pdg_{5153} = pdg_{1357}\)
  1. 3462972452; locally 8873965:
    \(v = v_0 + a t\)
    \(pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}\)
valid 4798787814:
3462972452:
4798787814:
3462972452:
5 declare final expr
  1. 3462972452; locally 8873965:
    \(v = v_0 + a t\)
    \(pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}\)
no validation is available for declarations 3462972452:
3462972452:
6 declare initial expr
  1. 3411994811; locally 8658331:
    \(v_{\rm average} = \frac{d}{t}\)
    \(pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}\)
no validation is available for declarations 3411994811:
3411994811:
7 declare initial expr
  1. 6175547907; locally 5013638:
    \(v_{\rm average} = \frac{v + v_0}{2}\)
    \(pdg_{6709} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\)
no validation is available for declarations 6175547907:
6175547907:
8 LHS of expr 1 equals LHS of expr 2
  1. 3411994811; locally 8658331:
    \(v_{\rm average} = \frac{d}{t}\)
    \(pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}\)
  2. 6175547907; locally 5013638:
    \(v_{\rm average} = \frac{v + v_0}{2}\)
    \(pdg_{6709} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\)
  1. 9897284307; locally 4622149:
    \(\frac{d}{t} = \frac{v + v_0}{2}\)
    \(\frac{pdg_{1943}}{pdg_{1467}} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\)
valid 3411994811:
6175547907:
9897284307:
3411994811:
6175547907:
9897284307:
9 multiply both sides by
  1. 9897284307; locally 4622149:
    \(\frac{d}{t} = \frac{v + v_0}{2}\)
    \(\frac{pdg_{1943}}{pdg_{1467}} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\)
  1. 8865085668:
    \(t\)
    \(pdg_{1467}\)
  1. 8706092970; locally 1476448:
    \(d = \left(\frac{v + v_0}{2}\right)t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)\)
valid 9897284307:
8706092970:
9897284307:
8706092970:
10 substitute RHS of expr 1 into expr 2
  1. 3462972452; locally 8873965:
    \(v = v_0 + a t\)
    \(pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}\)
  2. 8706092970; locally 1476448:
    \(d = \left(\frac{v + v_0}{2}\right)t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)\)
  1. 7011114072; locally 3069767:
    \(d = \frac{(v_0 + a t) + v_0}{2} t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)\)
LHS diff is 0 RHS diff is pdg1467*(pdg1357 - pdg1467*pdg9140 - pdg5153)/2 3462972452:
8706092970:
7011114072:
3462972452:
8706092970:
7011114072:
11 simplify
  1. 7011114072; locally 3069767:
    \(d = \frac{(v_0 + a t) + v_0}{2} t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)\)
  1. 1265150401; locally 6881977:
    \(d = \frac{2 v_0 + a t}{2} t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)\)
valid 7011114072:
1265150401:
7011114072:
1265150401:
12 simplify
  1. 1265150401; locally 6881977:
    \(d = \frac{2 v_0 + a t}{2} t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)\)
  1. 9658195023; locally 5385244:
    \(d = v_0 t + \frac{1}{2} a t^2\)
    \(pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}\)
valid 1265150401:
9658195023:
1265150401:
9658195023:
13 declare final expr
  1. 9658195023; locally 5385244:
    \(d = v_0 t + \frac{1}{2} a t^2\)
    \(pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}\)
no validation is available for declarations 9658195023:
9658195023:
14 raise both sides to power
  1. 3462972452; locally 8873965:
    \(v = v_0 + a t\)
    \(pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}\)
  1. 5799753649:
    \(2\)
    \(2\)
  1. 7215099603; locally 4385757:
    \(v^2 = v_0^2 + 2 a t v_0 + a^2 t^2\)
    \(pdg_{1357}^{2} = pdg_{1467}^{2} pdg_{9140}^{2} + 2 pdg_{1467} pdg_{5153} pdg_{9140} + pdg_{5153}^{2}\)
no check is performed 3462972452:
7215099603:
3462972452:
7215099603:
15 simplify
  1. 7215099603; locally 4385757:
    \(v^2 = v_0^2 + 2 a t v_0 + a^2 t^2\)
    \(pdg_{1357}^{2} = pdg_{1467}^{2} pdg_{9140}^{2} + 2 pdg_{1467} pdg_{5153} pdg_{9140} + pdg_{5153}^{2}\)
  1. 5144263777; locally 9796063:
    \(v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)\)
    \(pdg_{1357}\)
Nothing to split 7215099603:
5144263777:
7215099603:
5144263777:
factored 2a out of two terms
16 substitute RHS of expr 1 into expr 2
  1. 5144263777; locally 9796063:
    \(v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)\)
    \(pdg_{1357}\)
  2. 9658195023; locally 5385244:
    \(d = v_0 t + \frac{1}{2} a t^2\)
    \(pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}\)
  1. 7939765107; locally 7702534:
    \(v^2 = v_0^2 + 2 a d\)
    \(pdg_{1357}^{2} = 2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2}\)
Nothing to split 5144263777:
9658195023:
7939765107:
5144263777:
9658195023:
7939765107:
17 declare final expr
  1. 7939765107; locally 7702534:
    \(v^2 = v_0^2 + 2 a d\)
    \(pdg_{1357}^{2} = 2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2}\)
no validation is available for declarations 7939765107:
7939765107:
18 subtract X from both sides
  1. 3462972452; locally 8873965:
    \(v = v_0 + a t\)
    \(pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}\)
  1. 6729698807:
    \(v_0\)
    \(pdg_{5153}\)
  1. 9759901995; locally 4127918:
    \(v - v_0 = a t\)
    \(pdg_{1357} - pdg_{5153} = pdg_{1467} pdg_{9140}\)
valid 3462972452:
9759901995:
3462972452:
9759901995:
19 swap LHS with RHS
  1. 9759901995; locally 4127918:
    \(v - v_0 = a t\)
    \(pdg_{1357} - pdg_{5153} = pdg_{1467} pdg_{9140}\)
  1. 4748157455; locally 5666935:
    \(a t = v - v_0\)
    \(pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}\)
valid 9759901995:
4748157455:
9759901995:
4748157455:
20 divide both sides by
  1. 4748157455; locally 5666935:
    \(a t = v - v_0\)
    \(pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}\)
  1. 2242144313:
    \(a\)
    \(pdg_{9140}\)
  1. 1967582749; locally 8222540:
    \(t = \frac{v - v_0}{a}\)
    \(pdg_{1467} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{9140}}\)
valid 4748157455:
1967582749:
4748157455:
1967582749:
21 substitute RHS of expr 1 into expr 2
  1. 1967582749; locally 8222540:
    \(t = \frac{v - v_0}{a}\)
    \(pdg_{1467} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{9140}}\)
  2. 8706092970; locally 1476448:
    \(d = \left(\frac{v + v_0}{2}\right)t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)\)
  1. 5733721198; locally 9270356:
    \(d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)\)
    \(pdg_{1943} = \frac{\left(pdg_{1357} - pdg_{5153}\right) \left(pdg_{1357} + pdg_{5153}\right)}{2 pdg_{9140}}\)
LHS diff is 0 RHS diff is (pdg1357 + pdg5153)*(-pdg1357 + pdg1467*pdg9140 + pdg5153)/(2*pdg9140) 1967582749:
8706092970:
5733721198:
1967582749:
8706092970:
5733721198:
22 simplify
  1. 5733721198; locally 9270356:
    \(d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)\)
    \(pdg_{1943} = \frac{\left(pdg_{1357} - pdg_{5153}\right) \left(pdg_{1357} + pdg_{5153}\right)}{2 pdg_{9140}}\)
  1. 5611024898; locally 7103968:
    \(d = \frac{1}{2 a} (v^2 - v_0^2)\)
    \(pdg_{1943} = \frac{pdg_{1357}^{2} - pdg_{5153}^{2}}{2 pdg_{9140}}\)
valid 5733721198:
5611024898:
5733721198:
5611024898:
difference of squares
23 multiply both sides by
  1. 5611024898; locally 7103968:
    \(d = \frac{1}{2 a} (v^2 - v_0^2)\)
    \(pdg_{1943} = \frac{pdg_{1357}^{2} - pdg_{5153}^{2}}{2 pdg_{9140}}\)
  1. 5542390646:
    \(2 a\)
    \(2 pdg_{9140}\)
  1. 8269198922; locally 6814979:
    \(2 a d = v^2 - v_0^2\)
    \(2 pdg_{1943} pdg_{9140} = pdg_{1357}^{2} - pdg_{5153}^{2}\)
valid 5611024898:
8269198922:
5611024898:
8269198922:
24 add X to both sides
  1. 8269198922; locally 6814979:
    \(2 a d = v^2 - v_0^2\)
    \(2 pdg_{1943} pdg_{9140} = pdg_{1357}^{2} - pdg_{5153}^{2}\)
  1. 9070454719:
    \(v_0^2\)
    \(pdg_{5153}^{2}\)
  1. 4948763856; locally 7086842:
    \(2 a d + v_0^2 = v^2\)
    \(2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2} = pdg_{1357}^{2}\)
valid 8269198922:
4948763856:
8269198922:
4948763856:
25 swap LHS with RHS
  1. 4948763856; locally 7086842:
    \(2 a d + v_0^2 = v^2\)
    \(2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2} = pdg_{1357}^{2}\)
  1. 7939765107; locally 7702534:
    \(v^2 = v_0^2 + 2 a d\)
    \(pdg_{1357}^{2} = 2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2}\)
valid 4948763856:
7939765107:
4948763856:
7939765107:
26 declare final expr
  1. 8706092970; locally 1476448:
    \(d = \left(\frac{v + v_0}{2}\right)t\)
    \(pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)\)
no validation is available for declarations 8706092970:
8706092970:
27 subtract X from both sides
  1. 3462972452; locally 8873965:
    \(v = v_0 + a t\)
    \(pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}\)
  1. 9645178657:
    \(a t\)
    \(pdg_{1467} pdg_{9140}\)
  1. 6457044853; locally 8007427:
    \(v - a t = v_0\)
    \(pdg_{1357} - pdg_{1467} pdg_{9140} = pdg_{5153}\)
valid 3462972452:
6457044853:
3462972452:
6457044853:
28 substitute RHS of expr 1 into expr 2
  1. 6457044853; locally 8007427:
    \(v - a t = v_0\)
    \(pdg_{1357} - pdg_{1467} pdg_{9140} = pdg_{5153}\)
  2. 9658195023; locally 5385244:
    \(d = v_0 t + \frac{1}{2} a t^2\)
    \(pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}\)
  1. 1259826355; locally 5577530:
    \(d = (v - a t) t + \frac{1}{2} a t^2\)
    \(pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} \left(pdg_{1357} - pdg_{1467} pdg_{9140}\right)\)
valid 6457044853:
9658195023:
1259826355:
6457044853:
9658195023:
1259826355:
29 simplify
  1. 1259826355; locally 5577530:
    \(d = (v - a t) t + \frac{1}{2} a t^2\)
    \(pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} \left(pdg_{1357} - pdg_{1467} pdg_{9140}\right)\)
  1. 4580545876; locally 8442394:
    \(d = v t - a t^2 + \frac{1}{2} a t^2\)
    \(pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}\)
valid 1259826355:
4580545876:
1259826355:
4580545876:
30 simplify
  1. 4580545876; locally 8442394:
    \(d = v t - a t^2 + \frac{1}{2} a t^2\)
    \(pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}\)
  1. 6421241247; locally 3917794:
    \(d = v t - \frac{1}{2} a t^2\)
    \(pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}\)
valid 4580545876:
6421241247: dimensions are consistent
4580545876:
6421241247: N/A
31 declare final expr
  1. 6421241247; locally 3917794:
    \(d = v t - \frac{1}{2} a t^2\)
    \(pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}\)
no validation is available for declarations 6421241247: dimensions are consistent
6421241247: N/A
Physics Derivation Graph: Steps for equations of motion in 1D with constant acceleration - SUVAT (algebra)

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
9140 variable a
\(a\)
['real']
  • length: 1
  • time: -2
acceleration 29
1467 variable t
\(t\)
['real']
  • time: 1
time 115
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 80
6709 variable v_{\rm average}
\(v_{\rm average}\)
real
  • length: 1
  • time: -1
velocity average 2
1943 variable d
\(d\)
['real']
  • length: 1
displacement 25
5153 variable v_0
\(v_0\)
['real']
  • length: 1
  • time: -1
initial velocity 44
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