Projectile Motion Calculator
Calculate range, maximum height, time of flight, and final velocity of a projectile given initial velocity, launch angle, and optional initial height.
Results will appear here.
Formulas Used
Velocity Components:
v₀ₓ = v₀ · cos(θ)
v₀ᵧ = v₀ · sin(θ)
Time of Flight (from h₀ to ground, h = 0):
0 = h₀ + v₀ᵧ·t − ½·g·t²
t = (v₀ᵧ + √(v₀ᵧ² + 2·g·h₀)) / g
Horizontal Range:
R = v₀ₓ · t
Maximum Height:
H_max = h₀ + v₀ᵧ² / (2·g)
Time to Maximum Height:
t_max = v₀ᵧ / g
Final Speed at Impact:
v_f = √(v₀ₓ² + vᵧ_f²), where vᵧ_f = v₀ᵧ − g·t
Impact Angle Below Horizontal:
θ_impact = arctan(|vᵧ_f| / |v₀ₓ|)
Assumptions & References
- Motion occurs in a uniform gravitational field (constant g).
- Air resistance and wind effects are neglected.
- The projectile is treated as a point mass.
- The Earth's curvature and rotation are ignored.
- Launch and landing surfaces are flat (projectile lands at y = 0 unless h₀ > 0).
- Default gravitational acceleration: g = 9.81 m/s² (Earth surface).
- Reference: Halliday, Resnick & Krane, Physics, 5th ed., Chapter 4.
- Reference: Serway & Jewett, Physics for Scientists and Engineers, 8th ed.