Physics Derivation Graph navigation Sign in

review derivation: velocity at distance r of object dropped from infinity

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://www.youtube.com/watch?v=5F1XcTjpJs4 - Derivation of Gravitational Potential Energy by Rhett Allain

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
5 substitute LHS of expr 1 into expr 2
  1. 5902985919; locally 3470082:
    \(\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}\)
    \(\)
  2. 7882872592; locally 6798426:
    \(W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}\)
    \(\)
  1. 3566149658; locally 7300369:
    \(W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
    \(\)
failed 5902985919:
7882872592:
3566149658:
5902985919:
7882872592:
3566149658:
2 declare initial expr
  1. 5902985919; locally 3470082:
    \(\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}\)
    \(\)
no validation is available for declarations 5902985919:
5902985919:
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form
12 substitute LHS of expr 1 into expr 2
  1. 8049905441; locally 9781919:
    \(\Delta KE = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
  2. 1114820451; locally 9835406:
    \(W_{\rm by\ system} = \Delta KE\)
    \(\)
  1. 5779256336; locally 8118190:
    \(W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
valid 8049905441:
1114820451:
5779256336:
8049905441:
1114820451:
5779256336:
15 declare initial expr
  1. 2924222857; locally 1712972:
    \(v_{\rm initial} = v(r=\infty)\)
    \(\)
no validation is available for declarations 2924222857:
2924222857:
8 simplify
  1. 5596822289; locally 5818573:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
    \(\)
  1. 2061086175; locally 2429271:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
    \(\)
LHS diff is 0 RHS diff is pdg5022*pdg6277*(-pdg4851 + pdg4851(-1/pdg2530)) 5596822289:
2061086175:
5596822289:
2061086175:
11 declare initial expr
  1. 8357234146; locally 5104592:
    \(KE = \frac{1}{2} m v^2\)
    \(\)
no validation is available for declarations 8357234146:
8357234146:
26 declare final expr
  1. 2005061870; locally 3435796:
    \(v(r) = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
no validation is available for declarations 2005061870:
2005061870:
7 evaluate definite integral
  1. 8405272745; locally 9707318:
    \(W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
    \(\)
  1. 5596822289; locally 5818573:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
    \(\)
LHS diff is 0 RHS diff is pdg4851*pdg5022*pdg6277*(1 + 1/pdg2530) 8405272745:
5596822289:
8405272745:
5596822289:
9 simplify
  1. 2061086175; locally 2429271:
    \(W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
    \(\)
  1. 4393670960; locally 4947999:
    \(W_{\rm to\ system} = \frac{G m_1 m_2}{r}\)
    \(\)
LHS diff is 0 RHS diff is -pdg5022*pdg6277*(pdg2530*pdg4851(-1/pdg2530) + pdg4851)/pdg2530 2061086175:
4393670960:
2061086175:
4393670960:
25 change variable X to Y
  1. 5846639423; locally 7112224:
    \(v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
  1. 6599829782:
    \(v_{\rm final}\)
    \(\)
  2. 3531380618:
    \(v(r)\)
    \(\)
  1. 2005061870; locally 3435796:
    \(v(r) = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
valid 5846639423:
2005061870:
5846639423:
2005061870:
6 simplify
  1. 3566149658; locally 7300369:
    \(W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
    \(\)
  1. 8405272745; locally 9707318:
    \(W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
    \(\)
valid 3566149658:
8405272745:
3566149658:
8405272745:
16 substitute LHS of expr 1 into expr 2
  1. 3214170322; locally 8462685:
    \(v(r=\infty) = 0\)
    \(\)
  2. 2924222857; locally 1712972:
    \(v_{\rm initial} = v(r=\infty)\)
    \(\)
  1. 2998709778; locally 6923850:
    \(v_{\rm initial} = 0\)
    \(\)
Nothing to split 3214170322: no LHS/RHS split
2924222857:
2998709778:
3214170322: N/A
2924222857:
2998709778:
3 declare initial expr
  1. 1114820451; locally 9835406:
    \(W_{\rm by\ system} = \Delta KE\)
    \(\)
no validation is available for declarations 1114820451:
1114820451:
18 substitute LHS of expr 1 into expr 2
  1. 9510328252; locally 7110498:
    \(KE_{\rm initial} = 0\)
    \(\)
  2. 5779256336; locally 8118190:
    \(W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
  1. 5850144586; locally 2751634:
    \(W_{\rm by\ system} = KE_{\rm final}\)
    \(\)
valid 9510328252:
5779256336:
5850144586:
9510328252:
5779256336:
5850144586:
10 declare initial expr
  1. 8049905441; locally 9781919:
    \(\Delta KE = KE_{\rm final} - KE_{\rm initial}\)
    \(\)
no validation is available for declarations 8049905441:
8049905441:
13 change three variables in expr
  1. 8357234146; locally 5104592:
    \(KE = \frac{1}{2} m v^2\)
    \(\)
  1. 3731774096:
    \(KE\)
    \(\)
  2. 3350802342:
    \(KE_{\rm initial}\)
    \(\)
  3. 5904227750:
    \(m\)
    \(\)
  4. 6281834543:
    \(m_1\)
    \(\)
  5. 8066819515:
    \(v\)
    \(\)
  6. 3274176452:
    \(v_{\rm initial}\)
    \(\)
  1. 6091977310; locally 9031887:
    \(KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2\)
    \(\)
valid 8357234146:
6091977310:
8357234146:
6091977310:
21 substitute LHS of expr 1 into expr 2
  1. 9081138616; locally 6536576:
    \(W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
  2. 2907404069; locally 2619766:
    \(W_{\rm by\ system} = W_{\rm to\ system}\)
    \(\)
  1. 4947831649; locally 8655239:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}\)
    \(\)
valid 9081138616:
2907404069:
4947831649:
9081138616:
2907404069:
4947831649:
20 declare initial expr
  1. 2907404069; locally 2619766:
    \(W_{\rm by\ system} = W_{\rm to\ system}\)
    \(\)
no validation is available for declarations 2907404069:
2907404069:
22 substitute LHS of expr 1 into expr 2
  1. 4393670960; locally 4947999:
    \(W_{\rm to\ system} = \frac{G m_1 m_2}{r}\)
    \(\)
  2. 4947831649; locally 8655239:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}\)
    \(\)
  1. 6892595652; locally 2942416:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}\)
    \(\)
valid 4393670960:
4947831649:
6892595652:
4393670960:
4947831649:
6892595652:
17 substitute LHS of expr 1 into expr 2
  1. 2998709778; locally 6923850:
    \(v_{\rm initial} = 0\)
    \(\)
  2. 6091977310; locally 9031887:
    \(KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2\)
    \(\)
  1. 9510328252; locally 7110498:
    \(KE_{\rm initial} = 0\)
    \(\)
valid 2998709778:
6091977310:
9510328252:
2998709778:
6091977310:
9510328252:
14 change three variables in expr
  1. 8357234146; locally 5104592:
    \(KE = \frac{1}{2} m v^2\)
    \(\)
  1. 4587046017:
    \(KE\)
    \(\)
  2. 3939572542:
    \(KE_{\rm final}\)
    \(\)
  3. 9350720370:
    \(m\)
    \(\)
  4. 3166466250:
    \(m_1\)
    \(\)
  5. 6038673136:
    \(v\)
    \(\)
  6. 1616666229:
    \(v_{\rm final}\)
    \(\)
  1. 8552710882; locally 1397156:
    \(KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
failed 8357234146:
8552710882:
8357234146:
8552710882:
4 declare initial expr
  1. 7882872592; locally 6798426:
    \(W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}\)
    \(\)
no validation is available for declarations 7882872592:
7882872592:
24 square root both sides
  1. 7112646057; locally 4594601:
    \(v_{\rm final}^2 = \frac{2 G m_2}{r}\)
    \(\)
  1. 5846639423; locally 7112224:
    \(v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}\)
    \(\)
  2. 5693047217; locally 1366396:
    \(v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}\)
    \(\)
no check performed 7112646057:
5846639423:
5693047217:
7112646057:
5846639423:
5693047217:
19 substitute LHS of expr 1 into expr 2
  1. 8552710882; locally 1397156:
    \(KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
  2. 5850144586; locally 2751634:
    \(W_{\rm by\ system} = KE_{\rm final}\)
    \(\)
  1. 9081138616; locally 6536576:
    \(W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2\)
    \(\)
valid 8552710882:
5850144586:
9081138616:
8552710882:
5850144586:
9081138616:
23 multiply both sides by
  1. 6892595652; locally 2942416:
    \(\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}\)
    \(\)
  1. 7410526982:
    \(2/m_1\)
    \(\)
  1. 7112646057; locally 4594601:
    \(v_{\rm final}^2 = \frac{2 G m_2}{r}\)
    \(\)
valid 6892595652:
7112646057:
6892595652:
7112646057:
1 declare initial expr
  1. 3214170322; locally 8462685:
    \(v(r=\infty) = 0\)
    \(\)
no validation is available for declarations 3214170322: no LHS/RHS split
3214170322: N/A
starting velocity at infinity is zero
Physics Derivation Graph: Steps for velocity at distance r of object dropped from infinity

Symbols for this derivation

See also all 215 symbols
symbol ID category latex scope dimension name value Used in derivations references
4202 variable F
\(F\)
['real']
  • length: 1
  • mass: 1
  • time: -2
force 21
1934 variable v_{\rm initial}
\(v_{\rm initial}\)
real
  • length: 1
  • time: -1
initial velocity 4
4037 variable x
\(x\)
['real']
  • length: 1
position 53
6777 variable \vec{F}
\(\vec{F}\)
real dimensionless force 1
4121 variable KE_{\rm initial}
\(KE_{\rm initial}\)
real dimensionless initial kinetic energy 5
9372 variable W_{\rm to\ system}
\(W_{\rm to\ system}\)
real
  • length: 2
  • mass: 1
  • time: -2
work done to system 8
5340 variable KE_{\rm final}
\(KE_{\rm final}\)
real dimensionless final kinetic energy 5
4929 variable KE
\(KE\)
['real']
  • length: 2
  • mass: 1
  • time: -2
kinetic energy 7
5156 variable m
\(m\)
['real']
  • mass: 1
mass 57
6191 variable W_{\rm by\ system}
\(W_{\rm by\ system}\)
real
  • length: 2
  • mass: 1
  • time: -2
work done by system 5
5022 variable m_1
\(m_1\)
real
  • mass: 1
mass 18
4851 variable m_2
\(m_2\)
real
  • mass: 1
mass 17
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 80
5734 variable \Delta KE
\(\Delta KE\)
real dimensionless change in kinetic energy 2
2530 variable r
\(r\)
['real']
  • length: 1
radius 34
8909 variable v_{\rm final}
\(v_{\rm final}\)
real
  • length: 1
  • time: -1
final velocity 9
6277 constant G
\(G\)
real
  • length: 3
  • mass: -1
  • time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
46
MESSAGES: