Ballistic table based on launching from a height and angle with coasting ascent and descent (no drag, no thrust). Note the handy trigonometry identity for range: 2 sin θ cos θ = sin 2θ. This table is in pdf form here.
Spatio-temporal | Temporo-spatial |
Initial space angle = θ | Initial time angle = φ |
Initial height distance = y_{0} | Initial height distime = b_{0} |
Elapsed time interval = t | Elapsed stance interval = s |
Distance downrange or horizontal location = x | Distime downrange or horizontal chronation = a |
Altitude distance or vertical location = y | Altitude distime or vertical chronation = b |
Gravitational acceleration = g | Levitational relentation = h |
Initial velocity = v₀ | Initial lenticity = w₀ |
Initial horizontal velocity = v_{0x} = v_{0} cos θ | Initial horizontal lenticity = w_{0a} = w_{0} cos φ |
Initial vertical velocity = v_{0y} = v_{0} sin θ | Initial vertical lenticity = w_{0b} = w_{0} sin φ |
Horizontal velocity = v_{x} = v_{0x} | Horizontal lenticity = w_{a} = w_{0a} |
Vertical velocity = v_{y} = v_{0y} – gt | Vertical lenticity = w_{b} = w_{0b} – hs |
Velocity at apex point: v_{y} = 0 | Lenticity at apex instant: w_{b} = 0 |
Horizontal location x = v_{0x }t | Horizontal chronation a = w_{0a }s |
Vertical location y = v_{0y}t – ½ gt^{2} | Vertical chronation b = w_{0b}s – ½ hs^{2} |
Vertical location at impact point: y = 0 | Vertical chronation at impact instant: b = 0 |
Time of flight to apex t_{apex} = v_{0y}/g | Stance of flight to apex s_{apex} = w_{0b}/h |
Total time of flight t_{total} = 2t_{apex} = 2v_{0y}/g | Total stance of flight s_{total} = 2s_{apex} = 2w_{0b}/h |
Distance range to apex x_{apex} = v_{ox} v_{oy}/g | Distime range to apex a_{apex} = w_{oa} w_{ob}/h |
Total distance range x_{total} = 2v_{ox} v_{oy}/g | Total distime range a_{total} = 2w_{oa} w_{ob}/h |
Max altitude distance y_{apex} = ½ v_{0y}^{2}/g | Max altitude distime b_{apex} = ½ w_{0b}^{2}/h |
Trajectory formula: y = y_{0} + x tan θ − ½ gx²/v_{0x}^{2} | Trajectory formula: b = b_{0} + a tan φ − ½ ha²/w_{0a}^{2} |