Projectile Motion Calculator
Calculate 2D projectile trajectory range and height.
Quadratic Equation Solver (ax² + bx + c = 0)
About Projectile Motion Calculator
The 2D Projectile Motion & Ballistic Trajectory Calculator computes Total Flight Time ($T = \frac{2v_0\sin\theta}{g}$), Maximum Apex Height ($H = \frac{v_0^2\sin^2\theta}{2g}$), Horizontal Range ($R = \frac{v_0^2\sin(2\theta)}{g}$), Trajectory Parabola ($y(x)$), and impact angles with elevated launch platforms ($h_0 > 0$).
How to Use Projectile Motion Calculator
Step 1
Enter Launch Velocity ($v_0$), Launch Angle ($\theta$ in degrees), and optional Initial Elevation ($h_0$).
Step 2
Review computed range, maximum height, flight time, and parabolic flight plot.
Step 3
Click "Copy Trajectory Report".
Practical Use Cases for Projectile Motion Calculator
Sports Science & Ball Launch Angle Optimization (Golf, Baseball, Football)
Optimize launch angles and exit velocities to maximize flight carry distance and apex hang-time in sports analytics.
Military Ballistics & Fire Control Artillery Sizing
Compute target impact coordinates, flight time, and projectile trajectory curves from gun muzzle velocity and barrel elevation.
Input & Output Examples
Calculating Projectile Launch at 30 m/s at 45° Angle
Launch Velocity ($v_0$): `30.0 m/s`, Launch Angle ($ heta$): `45.0°`, Initial Height ($h_0$): `0.0 m`
Horizontal Range ($R$): `91.77 m` | Max Height ($H$): `22.94 m` | Flight Time ($T$): `4.33 s` | Impact Speed: `30.0 m/s at -45.0°`
Key Features & Performance
- ✓Computes: Range ($R$), Peak Apex Height ($H$), Total Hang Time ($T$), Horizontal/Vertical Velocity Components ($v_x, v_y$), Trajectory Equation.
- ✓Elevated Launch Support: handles launches from cliffs or elevated platforms ($h_0 > 0$) with exact quadratic roots.
- ✓Interactive Trajectory Canvas: visualizes the parabolic flight arc with draggable launch angles.
- ✓100% Client-Side ballistic engine.
- ✓1-Click Copy trajectory report.
Key Terminology & Definitions
Ballistic Trajectory
The curved parabolic flight path followed by an unpowered projectile under the action of gravity ($y(x) = h_0 + x\tan\theta - \frac{g x^2}{2v_0^2\cos^2\theta}$).
Optimal Launch Angle
For flat terrain ($h_0 = 0$), the angle producing maximum horizontal range in a vacuum is strictly 45 degrees ($\theta = 45^\circ$).