## review derivation: angle of maximum distance for projectile motion

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
7 divide both sides by
1. 1087417579; locally 7465542:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
1. 4829590294:
$$t_f$$
$$pdg_{2467}$$
1. 2086924031; locally 5115586:
$$0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}}{2} + pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
valid 1087417579:
2086924031:
1087417579:
2086924031:
4 LHS of expr 1 equals LHS of expr 2
1. 5379546684; locally 8592617:
$$y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$pdg_{7092} = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
2. 9112191201; locally 4911015:
$$y_f = 0$$
$$pdg_{7092} = 0$$
1. 8198310977; locally 7336772:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$0 = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
input diff is 0 diff is -pdg1469 + pdg1649*pdg2467**2/2 - pdg2467*pdg5153*sin(pdg1575) diff is pdg1469 - pdg1649*pdg2467**2/2 + pdg2467*pdg5153*sin(pdg1575) 5379546684:
9112191201:
8198310977:
5379546684:
9112191201:
8198310977:
22 declare final expr
1. 5353282496; locally 6972103:
$$d = \frac{v_0^2}{g}$$
$$pdg_{1943} = \frac{pdg_{5153}^{2}}{pdg_{1649}}$$
no validation is available for declarations 5353282496:
5353282496:
16 declare initial expr
1. 2405307372; locally 6199255:
$$\sin(2 x) = 2 \sin(x) \cos(x)$$
$$\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 2405307372:
2405307372:
10 declare initial expr
1. 5438722682; locally 2022953:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
no validation is available for declarations 5438722682:
5438722682:
21 simplify
1. 3607070319; locally 9834994:
$$d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(\frac{pdg_{3141}}{2} \right)}}{pdg_{1649}}$$
1. 5353282496; locally 6972103:
$$d = \frac{v_0^2}{g}$$
$$pdg_{1943} = \frac{pdg_{5153}^{2}}{pdg_{1649}}$$
LHS diff is 0 RHS diff is pdg5153**2*(sin(pdg3141/2) - 1)/pdg1649 3607070319:
5353282496:
3607070319:
5353282496:
12 boundary condition
1. 4370074654; locally 1654988:
$$t = t_f$$
$$pdg_{1467} = pdg_{2467}$$
1. 2378095808; locally 5891715:
$$x_f = x_0 + d$$
$$pdg_{3652} = pdg_{1572} + pdg_{1943}$$
no validation is available for assumptions 4370074654:
2378095808:
4370074654:
2378095808:
13 substitute LHS of expr 1 into expr 2
1. 2378095808; locally 5891715:
$$x_f = x_0 + d$$
$$pdg_{3652} = pdg_{1572} + pdg_{1943}$$
2. 3485125659; locally 2293278:
$$x_f = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{3652} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
1. 4268085801; locally 6742208:
$$x_0 + d = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{1572} + pdg_{1943} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
valid 2378095808:
3485125659:
4268085801:
2378095808:
3485125659:
4268085801:
14 subtract X from both sides
1. 4268085801; locally 6742208:
$$x_0 + d = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{1572} + pdg_{1943} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
1. 8072682558:
$$x_0$$
$$pdg_{1572}$$
1. 7233558441; locally 6756414:
$$d = v_0 t_f \cos(\theta)$$
$$pdg_{1943} = pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
valid 4268085801:
7233558441:
4268085801:
7233558441:
11 change two variables in expr
1. 5438722682; locally 2022953:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
1. 3273630811:
$$x$$
$$pdg_{4037}$$
2. 5194141542:
$$x_f$$
$$pdg_{3652}$$
3. 6732786762:
$$t$$
$$pdg_{1467}$$
4. 6463266449:
$$t_f$$
$$pdg_{2467}$$
1. 3485125659; locally 2293278:
$$x_f = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{3652} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
valid 5438722682:
3485125659:
5438722682:
3485125659:
2 change two variables in expr
1. 9862900242; locally 1292901:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
1. 8406170337:
$$y$$
$$pdg_{5647}$$
2. 8120663858:
$$y_f$$
$$pdg_{7092}$$
3. 2403773761:
$$t$$
$$pdg_{1467}$$
4. 4162188238:
$$t_f$$
$$pdg_{2467}$$
1. 5379546684; locally 8592617:
$$y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$pdg_{7092} = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
valid 9862900242:
5379546684:
9862900242:
5379546684:
6 substitute LHS of expr 1 into expr 2
1. 8198310977; locally 7336772:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$0 = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
2. 1650441634; locally 2601896:
$$y_0 = 0$$
$$pdg_{1469} = 0$$
1. 1087417579; locally 7465542:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
LHS diff is pdg1469 RHS diff is pdg1469 8198310977:
1650441634:
1087417579:
8198310977:
1650441634:
1087417579:
18 substitute LHS of expr 1 into expr 2
1. 2519058903; locally 7596368:
$$\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)$$
$$\sin{\left(2 pdg_{1575} \right)} = 2 \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}$$
2. 2297105551; locally 4362314:
$$d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)$$
$$pdg_{1943} = \frac{2 pdg_{5153}^{2} \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
1. 8922441655; locally 5129639:
$$d = \frac{v_0^2}{g} \sin(2 \theta)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(2 pdg_{1575} \right)}}{pdg_{1649}}$$
valid 2519058903:
2297105551:
8922441655: error for dim with 8922441655
2519058903:
2297105551:
8922441655: N/A
3 boundary condition
1. 5373931751; locally 7946350:
$$t = t_f$$
$$pdg_{1467} = pdg_{2467}$$
1. 9112191201; locally 4911015:
$$y_f = 0$$
$$pdg_{7092} = 0$$
no validation is available for assumptions 5373931751:
9112191201:
5373931751:
9112191201:
y(t_f) = y_f = 0
23 declare final expr
1. 1541916015; locally 2728170:
$$\theta = \frac{\pi}{4}$$
$$pdg_{1575} = \frac{pdg_{3141}}{4}$$
no validation is available for declarations 1541916015: dimensions are consistent
1541916015: N/A
9 multiply both sides by
1. 1191796961; locally 3904454:
$$\frac{1}{2} g t_f = v_0 \sin(\theta)$$
$$\frac{pdg_{1649} pdg_{2467}}{2} = pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
1. 2510804451:
$$2/g$$
$$\frac{2}{pdg_{1649}}$$
1. 4778077984; locally 8982886:
$$t_f = \frac{2 v_0 \sin(\theta)}{g}$$
$$pdg_{2467} = \frac{2 pdg_{5153} \sin{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
valid 1191796961:
4778077984:
1191796961:
4778077984:
15 substitute LHS of expr 1 into expr 2
1. 4778077984; locally 8982886:
$$t_f = \frac{2 v_0 \sin(\theta)}{g}$$
$$pdg_{2467} = \frac{2 pdg_{5153} \sin{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
2. 7233558441; locally 6756414:
$$d = v_0 t_f \cos(\theta)$$
$$pdg_{1943} = pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
1. 2297105551; locally 4362314:
$$d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)$$
$$pdg_{1943} = \frac{2 pdg_{5153}^{2} \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
valid 4778077984:
7233558441:
2297105551:
4778077984:
7233558441:
2297105551:
1 declare initial expr
1. 9862900242; locally 1292901:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
no validation is available for declarations 9862900242:
9862900242:
8 add X to both sides
1. 2086924031; locally 5115586:
$$0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}}{2} + pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
1. 6974054946:
$$\frac{1}{2} g t_f$$
$$\frac{pdg_{1649} pdg_{2467}}{2}$$
1. 1191796961; locally 3904454:
$$\frac{1}{2} g t_f = v_0 \sin(\theta)$$
$$\frac{pdg_{1649} pdg_{2467}}{2} = pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
valid 2086924031:
1191796961:
2086924031:
1191796961:
20 substitute LHS of expr 1 into expr 2
1. 1541916015; locally 2728170:
$$\theta = \frac{\pi}{4}$$
$$pdg_{1575} = \frac{pdg_{3141}}{4}$$
2. 8922441655; locally 5129639:
$$d = \frac{v_0^2}{g} \sin(2 \theta)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(2 pdg_{1575} \right)}}{pdg_{1649}}$$
1. 3607070319; locally 9834994:
$$d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(\frac{pdg_{3141}}{2} \right)}}{pdg_{1649}}$$
valid 1541916015: dimensions are consistent
8922441655: error for dim with 8922441655
3607070319:
1541916015: N/A
8922441655: N/A
3607070319:
5 declare assumption
1. 1650441634; locally 2601896:
$$y_0 = 0$$
$$pdg_{1469} = 0$$
no validation is available for declarations 1650441634:
1650441634:
17 change variable X to Y
1. 2405307372; locally 6199255:
$$\sin(2 x) = 2 \sin(x) \cos(x)$$
$$\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}$$
1. 7587034465:
$$\theta$$
$$pdg_{1575}$$
2. 7214442790:
$$x$$
$$pdg_{1464}$$
1. 2519058903; locally 7596368:
$$\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)$$
$$\sin{\left(2 pdg_{1575} \right)} = 2 \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}$$
LHS diff is sin(2*pdg1464) - sin(2*pdg1575) RHS diff is sin(2*pdg1464) - sin(2*pdg1575) 2405307372:
2519058903:
2405307372:
2519058903:
19 maximum of expr
1. 8922441655; locally 5129639:
$$d = \frac{v_0^2}{g} \sin(2 \theta)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(2 pdg_{1575} \right)}}{pdg_{1649}}$$
1. 5667870149:
$$\theta$$
$$pdg_{1575}$$
1. 1541916015; locally 2728170:
$$\theta = \frac{\pi}{4}$$
$$pdg_{1575} = \frac{pdg_{3141}}{4}$$
no check performed 8922441655: error for dim with 8922441655
1541916015: dimensions are consistent
8922441655: N/A
1541916015: N/A
Physics Derivation Graph: Steps for angle of maximum distance for projectile motion

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
$$x$$
['real'] dimensionless 140
2467 variable t_f
$$t_f$$
['real']
• time: 1
final time 15
1943 variable d
$$d$$
['real']
• length: 1
displacement
25
4037 variable x
$$x$$
['real']
• length: 1
position
53
1649 variable g
$$g$$
['real']
• length: 1
• time: -2
acceleration due to gravity
27
1467 variable t
$$t$$
['real']
• time: 1
time
121
7092 variable y_f
$$y_f$$
['real']
• length: 1
final position on y axis 3
1572 variable x_0
$$x_0$$
['real']
• length: 1
initial position 11
5153 variable v_0
$$v_0$$
['real']
• length: 1
• time: -1
initial velocity 44
5647 variable y
$$y$$
['real']
• length: 1
position
14
3652 variable x_f
$$x_f$$
real
• length: 1
final position on x axis 3
1469 variable y_0
$$y_0$$
['real']
• length: 1
initial position 9
3141 constant \pi
$$\pi$$
['real'] dimensionless pi 3.1415   dimensionless
72
1575 variable \theta
$$\theta$$
['real'] dimensionless angle
34
MESSAGE:
• local variable 'all_df' referenced before assignment