All calculatorsProjectile Motion Calculator
Launch a projectile and instantly get time of flight, range, maximum height, impact speed, and a mini trajectory visualization. Includes solve-for-speed and solve-for-angle target modes.
Background
Projectile motion (no air resistance) is just constant acceleration: horizontal velocity stays constant, while vertical motion accelerates downward by g. The path is a parabola.
How to use this calculator
- Pick a mode (Launch / Horizontal / Solve speed / Solve angle).
- Choose unit system, gravity g, and initial height y₀.
- Click Calculate for results + optional steps and a mini trajectory plot with peak and impact markers.
How this calculator works
- Horizontal: x-motion is constant; y-motion is free-fall.
- Launch: split velocity: v₀x=v₀cosθ, v₀y=v₀sinθ.
- Targets: uses the trajectory equation y(x) to solve for speed or angle when possible.
Formula & Equation Used
Components: v₀x=v₀cosθ, v₀y=v₀sinθ
Position vs time: x(t)=v₀x·t, y(t)=y₀+v₀y·t−(1/2)g t²
Trajectory: y(x)=y₀+x tanθ−(g x²)/(2 v₀² cos²θ)
Time to peak: tₚ=v₀y/g, Max height: yₘₐₓ=y₀+v₀y²/(2g)
Example Problem & Step-by-Step Solution
Example 1 — Classic 45° launch
- Given: v₀ = 20 m/s, θ = 45°, y₀ = 0, g = 9.8 m/s²
- Components: v₀x = v₀cosθ, v₀y = v₀sinθ
- Time of flight: solve y(t)=0 → get t_f
- Range: R = v₀x·t_f
Example 2 — Horizontal launch from a platform
- Given: vₓ = 9 m/s, y₀ = 12 m, g = 9.8 m/s²
- Fall time: t_f = √(2y₀/g)
- Range: R = vₓ·t_f
Example 3 — Solve angle(s) to hit a target
- Given: v₀ and target (x, y)
- Convert to quadratic in t = tanθ
- If the discriminant is positive, you can get two valid angles (low arc & high arc).
Frequently Asked Questions
Q: Why do I sometimes get two angles?
If the target is reachable at your speed, there’s often a low arc and a high arc that land at the same point.
Q: Why is my target “unreachable”?
The math requires a real solution (discriminant ≥ 0, and a positive denominator for v₀²). Try a different angle/speed or target height.
Q: Does this include air resistance?
No—this is the standard ideal projectile model (vacuum / no drag).
Q: If I switch to Imperial units, do the formulas change?
No—the physics is the same. We convert your inputs to SI internally, solve, then convert outputs back.
