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Ballistics

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 retardation = 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

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