With gravity as the only force, the horizontal and vertical motions of a projectile are independent. Horizontal velocity stays constant; vertical velocity decreases linearly under constant downward acceleration g.
Range: R = v02·sin(2θ) / g
Max height: H = v02·sin2(θ) / (2g)
Flight time:T = 2·v0·sin(θ) / g
Position: x(t) = v0·cos(θ)·t
y(t) = v0·sin(θ)·t − ½·g·t2
- Launch angle — 45° maximizes range in a vacuum; steeper angles trade range for height and hang time.
- Muzzle velocity — range and height both scale with v02, so doubling velocity quadruples them.
- Gravity — a weaker field (Moon, Mars) stretches the parabola dramatically; Jupiter's stronger pull flattens and shortens it.
- Air resistance — adds drag opposing velocity (∝ v2), curving the descent steeper than the ascent and shrinking range below the vacuum prediction — this is why real artillery tables never use the simple formulas above.