Horizontal position
x(t) = v₀ cosθ · t
Horizontal acceleration is zero in the ideal no-drag model.
Projectile Motion Calculator
Calculate ideal ballistic motion from launch speed, angle, initial height, and gravitational acceleration.
Access: Free to use, no installation, and No account required.
Inputs
This model neglects aerodynamic drag, lift, wind, spin, Earth curvature, and changing gravity. It assumes a flat landing surface at y = 0.
Results
Ideal projectile model
Horizontal velocity remains constant while vertical velocity changes at the entered gravitational acceleration. The calculation solves the time when the projectile returns to ground level.
Engineering reference
Projectile Motion Calculator: background and worked detail
Use the Projectile Motion Calculator to estimate ideal flight time, horizontal range, maximum height, launch velocity components, and impact speed from launch speed, angle, initial height, and constant gravity.
Separating horizontal and vertical motion
The model separates motion into constant horizontal velocity and uniformly accelerated vertical motion. With no aerodynamic drag, the horizontal and vertical components are coupled only through time.
The result is appropriate for textbook and first-order mechanics calculations. Air drag, lift, wind, spin, Earth curvature, variable gravity, terrain, and powered flight require a different trajectory model.
Equations used by this calculator
Vertical position
y(t) = y₀ + v₀ sinθ · t − ½gt²
Vertical acceleration is constant and downward.
Maximum height
y_max = y₀ + (v₀ sinθ)²/(2g)
For an upward launch, vertical velocity becomes zero at the apex.
30 m/s launch at 45° from ground level
Use speed 30 m/s, angle 45°, initial height 0 m, and g = 9.80665 m/s².
- Horizontal and vertical launch components are each about 21.21 m/s.
- Flight time is about 4.33 s.
- Range is about 91.8 m and maximum height about 22.9 m.
Result: With equal launch and landing height and no drag, the impact speed returns to 30 m/s.
No drag, no wind, flat ground
Assumptions
- Constant gravitational acceleration.
- No aerodynamic forces or wind.
- The ground intercept is the flat y = 0 plane.
Limitations
- Does not model drag, lift, spin, wind, Earth curvature, variable gravity, or terrain.
- At high speed, long range, or for real ballistic objects, aerodynamic and geodetic effects can dominate.
Validation checks
The 45° range maximum and symmetric trajectory give clean expected values.
Vertical launch
Case: Set angle to 90° at ground level.
Expected: Horizontal range should be approximately zero.
Energy symmetry
Case: Use equal launch and landing height with no drag.
Expected: Impact speed magnitude should equal launch speed within numerical rounding.
Projectile Motion Calculator FAQ
Is 45° always the maximum-range angle?
Only for the ideal no-drag case when launch and landing heights are equal. Drag or unequal elevations change the optimum angle.
Why is impact speed equal to launch speed in the default example?
With no drag and equal start/end height, mechanical energy is conserved, so the speed magnitude is the same.
Can I use this for a real projectile?
Use it as a first-order estimate. Real projectiles often need drag, wind, spin, density, and ballistic-coefficient modeling.
Where this calculation comes from
Shared with the Mechanics Workbench, which adds drag models and arbitrary launch geometry.
Open the source workbench →Read calculation and source methodology →