Projectile Motion Calculator
Enter initial velocity, launch angle and gravity to compute range, max height, flight time and velocity components of a projectile.
Input Data
Results
At a glance:Projectile motion (no air resistance) treats horizontal motion at constant velocity and vertical motion under constant gravity. Splitting v₀ into v_x = v₀·cosθ and v_y = v₀·sinθ: the horizontal velocity stays v_x, the vertical velocity changes as v_y − g·t. From these: range R = v₀²·sin(2θ)/g (launch and landing same height), max height H = v₀²·sin²θ/(2g), flight time T = 2v₀·sinθ/g, and vertical velocity at landing equals −v_y₀. Range is maximal at θ=45°. This tool returns R, H, T and the components from v₀, θ, g.
Formula
R = v₀²·sin(2θ)/g
H = v₀²·sin²θ/(2g)
T = 2v₀·sinθ/g
v_x = v₀·cosθ, v_y = v₀·sinθ
$$R = \frac{v_0^2 \sin(2\theta)}{g}, \quad H = \frac{v_0^2 \sin^2\theta}{2g}, \quad T = \frac{2v_0 \sin\theta}{g}$$How to Use
- Enter initial velocity v₀ (m/s) and launch angle θ (°).
- Enter gravity g (default 9.81).
- The calculator returns range, max height, flight time and velocity components.
Case Studies
Cannon at 45°
v₀ = 30 m/s, θ = 45°, g = 9.81.
R = 900×1/9.81 ≈ 91.7 m.
H = 900×0.5/19.62 ≈ 22.9 m, T ≈ 4.32 s.
FAQ
What angle gives maximum range?
θ = 45° (when launch and landing are at the same height), because sin(2θ) peaks at 1 there.
Why ignore air resistance?
It isolates the ideal parabolic trajectory and the clean formulas above; real projectiles fall shorter and lower due to drag (significant at high speed/long range).
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.