Angled-launch physics in a vacuum: range, max height, flight time, target-shooting angles, the inverse problem, and the 45° optimum — with an air-drag comparison and trajectory diagrams. Everything runs in your browser.
Frequently asked questions
What is the projectile motion formula?
For a projectile launched at speed v₀ and angle θ (launch and landing at the same height, no air resistance): range R = v₀²·sin(2θ)/g, maximum height H = (v₀·sinθ)²/(2g), and time of flight T = 2·v₀·sinθ/g. Example, verified: v₀ = 20 m/s at θ = 45° with g = 9.80665 m/s² gives R = 40.79 m, H = 10.20 m, T = 2.884 s. Try it in Mode 1 above — the calculator shows every step plus the trajectory diagram.
What launch angle gives the maximum range?
In a vacuum it is always exactly 45°, because sin(2θ) peaks at 2θ = 90°. The maximum range is then R_max = v₀²/g — for v₀ = 20 m/s on Earth that is 40.79 m. With air resistance the optimum drops below 45°: for a baseball (145 g, 7.4 cm, Cd 0.47) launched at 40 m/s, drag cuts the 45° range from 163.15 m to 83.92 m (a 48.6% loss) and the best angle shifts to about 40°. Use Mode 4 to run this comparison yourself.
How do you calculate the maximum height of a projectile?
H = (v₀·sinθ)²/(2g): only the vertical velocity component matters. A 20 m/s launch at 45° reaches H = 10.20 m; straight up (90°) it reaches v₀²/(2g) = 20.39 m. Note the apex is always exactly halfway in time (T/2) and, in a vacuum, halfway in range (R/2).
How do I find the launch angle to hit a specific target?
Write the trajectory as y = x·tanθ − (g·x²/(2v₀²))·(1+tan²θ). Substituting u = tanθ gives a quadratic with 0, 1, or 2 solutions: no solution means the target is out of range, one solution is the max-range grazing shot, two solutions are the low arc and the high arc. Example: v₀ = 20 m/s, target at x = 30 m, y = 5 m → θ = 36.16° (flat) or 63.30° (lobbed). Mode 2 solves this and shows the discriminant.
Can I work backwards from range and height to find the launch speed and angle?
Yes — for same-level launch and landing the two are linked by tanθ = 4H/R, then v₀ = √(R·g/sin(2θ)). Check: R = 40.79 m, H = 10.20 m gives θ = 45.01° and v₀ = 20.00 m/s, exactly round-tripping the Mode 1 example. Mode 3 does this inverse solve; it needs H > 0 (a perfectly flat H = 0 shot is degenerate).
What about a purely horizontal launch (θ = 0°)?
From ground level a 0° launch goes nowhere (R = 0, H = 0) — the interesting horizontal case is launching sideways from a height h, where t = √(2h/g) and R = v₀·t. That is a different tool: our sibling horizontal-projectile-motion covers horizontal launches from a height with impact velocity, impact angle, kinetic energy, and an inverse solver. Use this tool for angled (θ > 0°) launches.
Does air resistance really change the range that much?
Dramatically for light, fast objects. Mode 4 integrates quadratic drag (F = −k·|v|·v, k = ½·ρ·Cd·A) numerically and plots it against the vacuum trajectory. The baseball example above loses 48.6% of its range to drag. Heavy, slow, or streamlined projectiles lose far less — change the mass, diameter, and Cd inputs to see the sensitivity.
How is this different from a muzzle-velocity or ballistics calculator?
This is a school-physics vacuum model: constant g, launch/landing at one height, optional simple quadratic drag. Real ballistics (our sibling muzzle-velocity) works with grains, chronograph readings, ballistic coefficients, and velocity decay downrange — a different domain. Use this tool for physics homework, sports throws, and water-jet / sprinkler problems; use muzzle-velocity for firearms.
Can I use Moon or Mars gravity?
Yes — every mode has gravity presets: Earth 9.80665 m/s², Moon 1.62 m/s², Mars 3.71 m/s², plus a custom field. The same 20 m/s at 45° that flies 40.79 m on Earth flies 246.91 m on the Moon (T = 17.459 s) because range scales as 1/g.
What units does the calculator accept?
Launch speed in m/s, km/h, mph, or ft/s (converted internally to m/s); angles in degrees; distances out in metres with feet in parentheses. Reference: 20 m/s = 72 km/h = 44.74 mph = 65.62 ft/s, and 40.79 m = 133.82 ft. Gravity is always entered in m/s².