Physics Derivation Graph navigation Sign in

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

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

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

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

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

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
1467 variable t
\(t\)		 ['real']
  • time: 1
 time 121
5153 variable v_0
\(v_0\)		 ['real']
  • length: 1
  • time: -1
 initial velocity 44
1943 variable d
\(d\)		 ['real']
  • length: 1
 displacement 25
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
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
MESSAGE: