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 = y0 | Initial height distime = b0 |
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 = v0x = v0 cos θ | Initial horizontal lenticity = w0a = w0 cos φ |
Initial vertical velocity = v0y = v0 sin θ | Initial vertical lenticity = w0b = w0 sin φ |
Horizontal velocity = vx = v0x | Horizontal lenticity = wa = w0a |
Vertical velocity = vy = v0y – gt | Vertical lenticity = wb = w0b – hs |
Velocity at apex point: vy = 0 | Lenticity at apex instant: wb = 0 |
Horizontal location x = v0x t | Horizontal chronation a = w0a s |
Vertical location y = v0yt – ½ gt2 | Vertical chronation b = w0bs – ½ hs2 |
Vertical location at impact point: y = 0 | Vertical chronation at impact instant: b = 0 |
Time of flight to apex tapex = v0y/g | Stance of flight to apex sapex = w0b/h |
Total time of flight ttotal = 2tapex = 2v0y/g | Total stance of flight stotal = 2sapex = 2w0b/h |
Distance range to apex xapex = vox voy/g | Distime range to apex aapex = woa wob/h |
Total distance range xtotal = 2vox voy/g | Total distime range atotal = 2woa wob/h |
Max altitude distance yapex = ½ v0y2/g | Max altitude distime bapex = ½ w0b2/h |
Trajectory formula: y = y0 + x tan θ − ½ gx²/v0x2 | Trajectory formula: b = b0 + a tan φ − ½ ha²/w0a2 |