## 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
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:
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:
4580545876:
6421241247:
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:
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:
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:
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:
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:
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:
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:
6421241247:
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:
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:
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:
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:
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:
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:
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:
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: dimensions are consistent
6175547907:
9897284307:
3411994811: N/A
6175547907:
9897284307:
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:
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:
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:
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:
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
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:
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:
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:
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
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:
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:
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:
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: dimensions are consistent
3411994811: N/A
Physics Derivation Graph: Steps for equations of motion in 1D with constant acceleration - SUVAT (algebra)

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
1943 variable d
$$d$$
['real']
• length: 1
displacement
25
1357 variable v
$$v$$
['real']
• length: 1
• time: -1
velocity
83
1467 variable t
$$t$$
['real']
• time: 1
time
121
5153 variable v_0
$$v_0$$
['real']
• length: 1
• time: -1
initial velocity 44
6709 variable v_{\rm average}
$$v_{\rm average}$$
real
• length: 1
• time: -1
velocity average
2
9140 variable a
$$a$$
['real']
• length: 1
• time: -2
acceleration 31
MESSAGE:
• local variable 'all_df' referenced before assignment