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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:
    \(-g t + v_{0, y} = \frac{dy}{dt}\)
    \(- pdg_{1467} pdg_{6277} + pdg_{9431} = \frac{d}{d pdg_{1467}} pdg_{5647}\)
LHS diff is pdg1467*(-pdg1649 + pdg6277) 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: inconsistent dimensions
 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:
    \(-g t + v_{0, y} = \frac{dy}{dt}\)
    \(- pdg_{1467} pdg_{6277} + pdg_{9431} = \frac{d}{d pdg_{1467}} pdg_{5647}\)
  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*(pdg1649 - pdg6277) 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: inconsistent dimensions
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

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
2423 variable \vec{a}
\(\vec{a}\)		 real
  • length: 1
  • time: -2
 acceleration
  • str_note
2
1469 variable y_0
\(y_0\)		 ['real']
  • length: 1
 initial position 9
9199 variable dx
\(dx\)		 ['real']
  • length: 1
 15
1467 variable t
\(t\)		 ['real']
  • time: 1
 time 121
2958 variable v_{0, x}
\(v_{0, x}\)		 ['real']
  • length: 1
  • time: -1
 initial velocity along x axis 15
9431 variable v_{0, y}
\(v_{0, y}\)		 ['real']
  • length: 1
  • time: -1
 initial velocity along y axis 12
8339 variable \hat{x}
\(\hat{x}\)		 real dimensionless unit vector 4
6091 variable \vec{v}_0
\(\vec{v}_0\)		 ['real']
  • length: 1
  • time: -1
 initial velocity 1
5842 variable dy
\(dy\)		 ['real']
  • length: 1
 differential displacement along y axis
  • str_note
2
5505 variable v_x
\(v_x\)		 real
  • length: 1
  • time: -1
 velocity along x axis
  • str_note
7
7159 variable a_x
\(a_x\)		 real
  • length: 1
  • time: -2
 acceleration along x axis 4
5005 variable d v_x
\(d v_x\)		 ['real']
  • length: 1
  • time: -1
 differential velocity along x axis 2
1649 variable g
\(g\)		 ['real']
  • length: 1
  • time: -2
 acceleration due to gravity 27
5674 variable d v_y
\(d v_y\)		 ['real']
  • length: 1
  • time: -1
 differential velocity along y axis 2
1575 variable \theta
\(\theta\)		 ['real'] dimensionless angle 34
5647 variable y
\(y\)		 ['real']
  • length: 1
 position 14
6373 variable \vec{v}
\(\vec{v}\)		 real
  • length: 1
  • time: -1
 velocity
  • str_note
2
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
5153 variable v_0
\(v_0\)		 ['real']
  • length: 1
  • time: -1
 initial velocity 44
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 60
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
4711 variable dt
\(dt\)		 ['real']
  • time: 1
 differential time
  • str_note
7
1464 variable x
\(x\)		 ['real'] dimensionless 140
1700 variable \hat{y}
\(\hat{y}\)		 real dimensionless unit vector 4
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