## review derivation: equations of motion in 2D (calculus)

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
28 substitute LHS of expr 1 into expr 2
1. 6134836751; locally 8435615:
$$v_{0, x} = v_x$$
$$pdg_{2958} = pdg_{5505}$$
2. 8460820419; locally 4895553:
$$v_x = \frac{dx}{dt}$$
$$pdg_{5505} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
1. 7455581657; locally 5123314:
$$v_{0, x} = \frac{dx}{dt}$$
$$pdg_{2958} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
LHS diff is -pdg2958 + pdg5505 RHS diff is 0 6134836751: dimensions are consistent
8460820419:
7455581657:
6134836751: N/A
8460820419:
7455581657:
17 declare initial expr
1. 7252338326; locally 3936380:
$$v_y = \frac{dy}{dt}$$
$$pdg_{9107} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
no validation is available for declarations 7252338326:
7252338326:
23 multiply both sides by
1. 8750379055; locally 8742281:
$$0 = \frac{d}{dt} v_x$$
$$0 = \frac{d}{d pdg_{1467}} pdg_{5505}$$
1. 8717193282:
$$dt$$
$$pdg_{4711}$$
1. 1166310428; locally 5239397:
$$0 dt = d v_x$$
$$0 = pdg_{5005}$$
LHS diff is 0 RHS diff is -pdg5005 8750379055:
1166310428:
8750379055:
1166310428:
10 assume N dimensions
1. 8880467139:
$$2$$
$$2$$
1. 5349866551; locally 5359560:
$$\vec{v} = v_x \hat{x} + v_y \hat{y}$$
$$pdg_{6373} = pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}$$
no validation is available for assumptions 5349866551:
5349866551:
29 multiply both sides by
1. 7455581657; locally 5123314:
$$v_{0, x} = \frac{dx}{dt}$$
$$pdg_{2958} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
1. 8607458157:
$$dt$$
$$pdg_{4711}$$
1. 1963253044; locally 8062944:
$$v_{0, x} dt = dx$$
$$pdg_{2958} pdg_{4711} = pdg_{9199}$$
LHS diff is 0 RHS diff is -pdg9199 7455581657:
1963253044:
7455581657:
1963253044:
16 add X to both sides
1. 9973952056; locally 1321587:
$$-g t = v_y - v_{0, y}$$
$$- pdg_{1467} pdg_{1649} = - pdg_{5153} + pdg_{9431}$$
1. 4167526462:
$$v_{0, y}$$
$$pdg_{9431}$$
1. 6572039835; locally 2682139:
$$-g t + v_{0, y} = v_y$$
$$- pdg_{1467} pdg_{1649} + pdg_{9431} = pdg_{9107}$$
LHS diff is 0 RHS diff is -pdg5153 - pdg9107 + 2*pdg9431 9973952056:
6572039835:
9973952056:
6572039835:
11 substitute LHS of expr 1 into expr 2
1. 9707028061; locally 2060958:
$$a_x = 0$$
$$pdg_{7159} = 0$$
2. 1819663717; locally 5765841:
$$a_x = \frac{d}{dt} v_x$$
$$pdg_{7159} = \frac{d}{d pdg_{1467}} pdg_{5505}$$
1. 8750379055; locally 8742281:
$$0 = \frac{d}{dt} v_x$$
$$0 = \frac{d}{d pdg_{1467}} pdg_{5505}$$
valid 9707028061:
1819663717:
8750379055:
9707028061:
1819663717:
8750379055:
30 indefinite integration
1. 1963253044; locally 8062944:
$$v_{0, x} dt = dx$$
$$pdg_{2958} pdg_{4711} = pdg_{9199}$$
1. 3676159007; locally 2732393:
$$v_{0, x} \int dt = \int dx$$
$$pdg_{2958} \int 1\, dpdg_{1467} = \int 1\, dpdg_{1464}$$
no check performed 1963253044:
3676159007:
1963253044:
3676159007:
39 multiply both sides by
1. 7376526845; locally 2378061:
$$\sin(\theta) = \frac{v_{0, y}}{v_0}$$
$$\sin{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{9431}}$$
1. 5620558729:
$$v_0$$
$$pdg_{5153}$$
1. 8949329361; locally 3041148:
$$v_0 \sin(\theta) = v_{0, y}$$
$$pdg_{5153} \sin{\left(pdg_{1575} \right)} = pdg_{9431}$$
LHS diff is 0 RHS diff is pdg5153**2/pdg9431 - pdg9431 7376526845:
8949329361:
7376526845:
8949329361:
33 swap LHS with RHS
1. 8486706976; locally 6277762:
$$v_{0, x} t + x_0 = x$$
$$pdg_{1467} pdg_{2958} + pdg_{1572} = pdg_{4037}$$
1. 1306360899; locally 3011802:
$$x = v_{0, x} t + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{2958} + pdg_{1572}$$
valid 8486706976:
1306360899:
8486706976:
1306360899:
12 substitute LHS of expr 1 into expr 2
1. 2741489181; locally 1439312:
$$a_y = -g$$
$$pdg_{7055} = - pdg_{1649}$$
2. 8228733125; locally 2080932:
$$a_y = \frac{d}{dt} v_y$$
$$pdg_{7055} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
1. 1977955751; locally 3939933:
$$-g = \frac{d}{dt} v_y$$
$$- pdg_{1649} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
valid 2741489181:
8228733125:
1977955751:
2741489181:
8228733125:
1977955751:
6 separate two vector components
1. 7729413831; locally 4904941:
$$a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)$$
$$pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339} = \frac{\partial}{\partial pdg_{1467}} \left(pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}\right)$$
1. 1819663717; locally 5765841:
$$a_x = \frac{d}{dt} v_x$$
$$pdg_{7159} = \frac{d}{d pdg_{1467}} pdg_{5505}$$
2. 8228733125; locally 2080932:
$$a_y = \frac{d}{dt} v_y$$
$$pdg_{7055} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
no check performed 7729413831:
1819663717:
8228733125:
7729413831:
1819663717:
8228733125:
13 multiply both sides by
1. 1977955751; locally 3939933:
$$-g = \frac{d}{dt} v_y$$
$$- pdg_{1649} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
1. 6672141531:
$$dt$$
$$pdg_{4711}$$
1. 1702349646; locally 4777195:
$$-g dt = d v_y$$
$$- dt pdg_{1649} = pdg_{5674}$$
LHS diff is pdg1649*(dt - pdg4711) RHS diff is -pdg5674 1977955751:
1702349646:
1977955751:
1702349646:
37 substitute LHS of expr 1 into expr 2
1. 6083821265; locally 6010171:
$$v_0 \cos(\theta) = v_{0, x}$$
$$pdg_{5153} \cos{\left(pdg_{1575} \right)} = pdg_{2958}$$
2. 1306360899; locally 3011802:
$$x = v_{0, x} t + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{2958} + pdg_{1572}$$
1. 5438722682; locally 6795282:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
LHS diff is 0 RHS diff is pdg1467*(pdg2958 - pdg5153*cos(pdg1575)) 6083821265:
1306360899:
5438722682:
6083821265:
1306360899:
5438722682:
14 indefinite integration
1. 1702349646; locally 4777195:
$$-g dt = d v_y$$
$$- dt pdg_{1649} = pdg_{5674}$$
1. 8584698994; locally 3366698:
$$-g \int dt = \int d v_y$$
$$- dt g = pdg_{5674}$$
no check performed 1702349646:
8584698994:
1702349646:
8584698994:
40 swap LHS with RHS
1. 2461349007; locally 7541692:
$$- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} + pdg_{1469} = pdg_{5647}$$
1. 1405465835; locally 1910429:
$$y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9107} + pdg_{1469}$$
LHS diff is pdg1467*(-pdg9107 + pdg9431) RHS diff is pdg1467*(-pdg9107 + pdg9431) 2461349007:
1405465835:
2461349007:
1405465835:
22 add X to both sides
1. 2858549874; locally 8638087:
$$- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} = - pdg_{1469} + pdg_{5647}$$
1. 6098638221:
$$y_0$$
$$pdg_{1469}$$
1. 2461349007; locally 7541692:
$$- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} + pdg_{1469} = pdg_{5647}$$
valid 2858549874:
2461349007:
2858549874:
2461349007:
15 simplify
1. 8584698994; locally 3366698:
$$-g \int dt = \int d v_y$$
$$- dt g = pdg_{5674}$$
1. 9973952056; locally 1321587:
$$-g t = v_y - v_{0, y}$$
$$- pdg_{1467} pdg_{1649} = - pdg_{5153} + pdg_{9431}$$
LHS diff is -dt*g + pdg1467*pdg1649 RHS diff is pdg5153 + pdg5674 - pdg9431 8584698994:
9973952056:
8584698994:
9973952056:
7 declare assumption
1. 9707028061; locally 2060958:
$$a_x = 0$$
$$pdg_{7159} = 0$$
no validation is available for declarations 9707028061:
9707028061:
define the orientation of the coordinate system with respect to the gravitational acceleration such that x axis is perpendicular to gravity
8 declare assumption
1. 2741489181; locally 1439312:
$$a_y = -g$$
$$pdg_{7055} = - pdg_{1649}$$
no validation is available for declarations 2741489181:
2741489181:
define the orientation of the coordinate system with respect to the gravitational acceleration such that y axis is parallel to gravity
36 multiply both sides by
1. 7391837535; locally 5523081:
$$\cos(\theta) = \frac{v_{0, x}}{v_0}$$
$$\cos{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{2958}}$$
1. 5868731041:
$$v_0$$
$$pdg_{5153}$$
1. 6083821265; locally 6010171:
$$v_0 \cos(\theta) = v_{0, x}$$
$$pdg_{5153} \cos{\left(pdg_{1575} \right)} = pdg_{2958}$$
LHS diff is 0 RHS diff is -pdg2958 + pdg5153**2/pdg2958 7391837535:
6083821265:
7391837535:
6083821265:
32 add X to both sides
1. 9882526611; locally 2740672:
$$v_{0, x} t = x - x_0$$
$$pdg_{1467} pdg_{2958} = - pdg_{1572} + pdg_{4037}$$
1. 3182907803:
$$x_0$$
$$pdg_{1572}$$
1. 8486706976; locally 6277762:
$$v_{0, x} t + x_0 = x$$
$$pdg_{1467} pdg_{2958} + pdg_{1572} = pdg_{4037}$$
valid 9882526611:
8486706976:
9882526611:
8486706976:
5 substitute LHS of expr 1 into expr 2
1. 5349866551; locally 5359560:
$$\vec{v} = v_x \hat{x} + v_y \hat{y}$$
$$pdg_{6373} = pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}$$
2. 4158986868; locally 4755350:
$$a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}$$
$$pdg_{1467}$$
1. 7729413831; locally 4904941:
$$a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)$$
$$pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339} = \frac{\partial}{\partial pdg_{1467}} \left(pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}\right)$$
Nothing to split 5349866551:
4158986868:
7729413831:
5349866551:
4158986868:
7729413831:
24 indefinite integration
1. 1166310428; locally 5239397:
$$0 dt = d v_x$$
$$0 = pdg_{5005}$$
1. 2366691988; locally 3137944:
$$\int 0 dt = \int d v_x$$
$$\int 0\, dpdg_{1467} = \int 1\, dpdg_{5005}$$
no check performed 1166310428:
2366691988:
1166310428:
2366691988:
42 declare final expr
1. 9862900242; locally 9780510:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
no validation is available for declarations 9862900242:
9862900242:
38 declare final expr
1. 5438722682; locally 6795282:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
no validation is available for declarations 5438722682:
5438722682:
9 assume N dimensions
1. 3270039798:
$$2$$
$$2$$
1. 8602512487; locally 4862823:
$$\vec{a} = a_x \hat{x} + a_y \hat{y}$$
$$pdg_{2423} = pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339}$$
no validation is available for assumptions 8602512487:
8602512487:
18 substitute LHS of expr 1 into expr 2
1. 7252338326; locally 3936380:
$$v_y = \frac{dy}{dt}$$
$$pdg_{9107} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
2. 6572039835; locally 2682139:
$$-g t + v_{0, y} = v_y$$
$$- pdg_{1467} pdg_{1649} + pdg_{9431} = pdg_{9107}$$
1. 6204539227; locally 5010170:
$$-g t + v_{0, y} = \frac{dy}{dt}$$
$$- g pdg_{1467} + pdg_{9431} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
LHS diff is pdg1467*(g - pdg1649) RHS diff is 0 7252338326:
6572039835:
6204539227:
7252338326:
6572039835:
6204539227:
27 declare initial expr
1. 8460820419; locally 4895553:
$$v_x = \frac{dx}{dt}$$
$$pdg_{5505} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
no validation is available for declarations 8460820419:
8460820419:
25 simplify
1. 2366691988; locally 3137944:
$$\int 0 dt = \int d v_x$$
$$\int 0\, dpdg_{1467} = \int 1\, dpdg_{5005}$$
1. 1676472948; locally 9737190:
$$0 = v_x - v_{0, x}$$
$$0 = - pdg_{2958} + pdg_{5505}$$
LHS diff is 0 RHS diff is pdg2958 + pdg5005 - pdg5505 2366691988:
1676472948: error for dim with 1676472948
2366691988:
1676472948: N/A
3 substitute LHS of expr 1 into expr 2
1. 3169580383; locally 6758737:
$$\vec{a} = \frac{d\vec{v}}{dt}$$
$$pdg_{2423} = \frac{d}{d pdg_{1467}} pdg_{6373}$$
2. 8602512487; locally 4862823:
$$\vec{a} = a_x \hat{x} + a_y \hat{y}$$
$$pdg_{2423} = pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339}$$
1. 4158986868; locally 4755350:
$$a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}$$
$$pdg_{1467}$$
Nothing to split 3169580383:
8602512487:
4158986868:
3169580383:
8602512487:
4158986868:
20 indefinite integration
1. 8145337879; locally 5577963:
$$-g t dt + v_{0, y} dt = dy$$
$$- pdg_{1467} pdg_{1649} pdg_{4711} + pdg_{4711} pdg_{9431} = pdg_{5842}$$
1. 8808860551; locally 8020644:
$$-g \int t dt + v_{0, y} \int dt = \int dy$$
$$- pdg_{1649} \int pdg_{1467}\, dpdg_{1467} + pdg_{9431} \int 1\, dpdg_{1467} = \int 1\, dpdg_{5647}$$
no check performed 8145337879:
8808860551:
8145337879:
8808860551:
35 separate vector into two trigonometric ratios
1. 9341391925; locally 1381925:
$$\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}$$
$$pdg_{6091} = pdg_{1700} pdg_{9431} + pdg_{2958} pdg_{8339}$$
1. 6410818363:
$$\theta$$
$$pdg_{1575}$$
1. 7391837535; locally 5523081:
$$\cos(\theta) = \frac{v_{0, x}}{v_0}$$
$$\cos{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{2958}}$$
2. 7376526845; locally 2378061:
$$\sin(\theta) = \frac{v_{0, y}}{v_0}$$
$$\sin{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{9431}}$$
no check performed 9341391925:
7391837535:
7376526845:
9341391925:
7391837535:
7376526845:
41 substitute LHS of expr 1 into expr 2
1. 8949329361; locally 3041148:
$$v_0 \sin(\theta) = v_{0, y}$$
$$pdg_{5153} \sin{\left(pdg_{1575} \right)} = pdg_{9431}$$
2. 1405465835; locally 1910429:
$$y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9107} + pdg_{1469}$$
1. 9862900242; locally 9780510:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
LHS diff is 0 RHS diff is pdg1467*(-pdg5153*sin(pdg1575) + pdg9107) 8949329361:
1405465835:
9862900242:
8949329361:
1405465835:
9862900242:
21 simplify
1. 8808860551; locally 8020644:
$$-g \int t dt + v_{0, y} \int dt = \int dy$$
$$- pdg_{1649} \int pdg_{1467}\, dpdg_{1467} + pdg_{9431} \int 1\, dpdg_{1467} = \int 1\, dpdg_{5647}$$
1. 2858549874; locally 8638087:
$$- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} = - pdg_{1469} + pdg_{5647}$$
LHS diff is 0 RHS diff is pdg1469 8808860551:
2858549874:
8808860551:
2858549874:
31 simplify
1. 3676159007; locally 2732393:
$$v_{0, x} \int dt = \int dx$$
$$pdg_{2958} \int 1\, dpdg_{1467} = \int 1\, dpdg_{1464}$$
1. 9882526611; locally 2740672:
$$v_{0, x} t = x - x_0$$
$$pdg_{1467} pdg_{2958} = - pdg_{1572} + pdg_{4037}$$
LHS diff is 0 RHS diff is pdg1464 + pdg1572 - pdg4037 3676159007:
9882526611:
3676159007:
9882526611:
34 assume N dimensions
1. 7049769409:
$$2$$
$$2$$
1. 9341391925; locally 1381925:
$$\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}$$
$$pdg_{6091} = pdg_{1700} pdg_{9431} + pdg_{2958} pdg_{8339}$$
no validation is available for assumptions 9341391925:
9341391925:
19 multiply both sides by
1. 6204539227; locally 5010170:
$$-g t + v_{0, y} = \frac{dy}{dt}$$
$$- g pdg_{1467} + pdg_{9431} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
1. 1614343171:
$$dt$$
$$pdg_{4711}$$
1. 8145337879; locally 5577963:
$$-g t dt + v_{0, y} dt = dy$$
$$- pdg_{1467} pdg_{1649} pdg_{4711} + pdg_{4711} pdg_{9431} = pdg_{5842}$$
LHS diff is pdg1467*pdg4711*(-g + pdg1649) RHS diff is -pdg5842 6204539227:
8145337879:
6204539227:
8145337879:
1 declare initial expr
1. 3169580383; locally 6758737:
$$\vec{a} = \frac{d\vec{v}}{dt}$$
$$pdg_{2423} = \frac{d}{d pdg_{1467}} pdg_{6373}$$
no validation is available for declarations 3169580383:
3169580383:
26 add X to both sides
1. 1676472948; locally 9737190:
$$0 = v_x - v_{0, x}$$
$$0 = - pdg_{2958} + pdg_{5505}$$
1. 1439089569:
$$v_{0, x}$$
$$pdg_{2958}$$
1. 6134836751; locally 8435615:
$$v_{0, x} = v_x$$
$$pdg_{2958} = pdg_{5505}$$
valid 1676472948: error for dim with 1676472948
6134836751: dimensions are consistent
1676472948: N/A
6134836751: N/A
Physics Derivation Graph: Steps for equations of motion in 2D (calculus)

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
1700 variable \hat{y}
$$\hat{y}$$
real dimensionless unit vector
4
5005 variable d v_x
$$d v_x$$
['real']
• length: 1
• time: -1
differential velocity along x axis 2
5842 variable dy
$$dy$$
['real']
• length: 1
differential displacement along y axis
• str_note
2
5153 variable v_0
$$v_0$$
['real']
• length: 1
• time: -1
initial velocity 44
1469 variable y_0
$$y_0$$
['real']
• length: 1
initial position 9
9431 variable v_{0, y}
$$v_{0, y}$$
['real']
• length: 1
• time: -1
initial velocity along y axis
12
4711 variable dt
$$dt$$
['real']
• time: 1
differential time
• str_note
7
4037 variable x
$$x$$
['real']
• length: 1
position
53
1575 variable \theta
$$\theta$$
['real'] dimensionless angle
34
5647 variable y
$$y$$
['real']
• length: 1
position
14
1467 variable t
$$t$$
['real']
• time: 1
time
120
2423 variable \vec{a}
$$\vec{a}$$
real
• length: 1
• time: -2
acceleration
• str_note
2
7159 variable a_x
$$a_x$$
real
• length: 1
• time: -2
acceleration along x axis
4
6373 variable \vec{v}
$$\vec{v}$$
real
• length: 1
• time: -1
velocity
• str_note
2
8339 variable \hat{x}
$$\hat{x}$$
real dimensionless unit vector
4
5674 variable d v_y
$$d v_y$$
['real']
• length: 1
• time: -1
differential velocity along y axis 2
2958 variable v_{0, x}
$$v_{0, x}$$
['real']
• length: 1
• time: -1
initial velocity along x axis
15
1572 variable x_0
$$x_0$$
['real']
• length: 1
initial position 11
9107 variable v_y
$$v_y$$
real
• length: 1
• time: -1
velocity along y axis
• str_note
7
7055 variable a_y
$$a_y$$
real
• length: 1
• time: -2
acceleration along y axis
4
1464 variable x
$$x$$
['real'] dimensionless 140
5505 variable v_x
$$v_x$$
real
• length: 1
• time: -1
velocity along x axis
• str_note
7
9199 variable dx
$$dx$$
['real']
• length: 1
15
1649 variable g
$$g$$
['real']
• length: 1
• time: -2
acceleration due to gravity
27
6091 variable \vec{v}_0
$$\vec{v}_0$$
['real']
• length: 1
• time: -1
initial velocity
1
MESSAGE:
• local variable 'all_df' referenced before assignment