Projectile Motion & Ballistic Trajectory Calculator

1. The "45 Degrees is Always Maximum Range" Fallacy High school physics students memorize that 45° produces maximum range. This is true ONLY when launch height equals landing height ($y_0 = 0$). When throwing from a cliff, standing height, or aircraft ($y_0 > 0$), the optimal angle is strictly less than 45°: $\theta = \arcsin(1 / \sqrt{2 + 2gy_0/v_0^2})$. In shot put (release height 2.1m), optimal launch angle is roughly 37°–38°.

2. Neglecting Aerodynamic Drag in Real-World Athletics In baseball, golf, or artillery, atmospheric drag forces ($F_d = \frac{1}{2}\rho v^2 C_d A$) alter trajectories radically. A 100 mph baseball hit in a vacuum travels over 700 feet; with air drag and Magnus spin backspin, it travels roughly 400 feet and exhibits an asymmetric trajectory with a steep, abrupt terminal drop.

3. JavaScript Radians vs. Degrees Calculation Bug JavaScript math functions (Math.sin(), Math.cos()) strictly expect angles in radians. Passing raw degrees without converting $ heta_{ ext{rad}} = heta_{ ext{deg}} imes (pi / 180)$ generates nonsensical negative flight times and negative ranges.

4. Overlooking Kinetic Energy Transformation at Apex At the trajectory apex, vertical velocity reaches zero ($v_y = 0$), but kinetic energy is NOT zero. The projectile still retains 100% of its horizontal kinetic energy: $E_k = \frac{1}{2}m v_x^2 = \frac{1}{2}m (v_0 \cos\theta)^2$. Only in a purely vertical 90° launch does kinetic energy drop to absolute zero at peak.

5. Coriolis Acceleration in Long-Range Ballistics (>1,000m) For extreme long-range rifle shooting and naval artillery, Earth rotates beneath the projectile during its multi-second flight. In the Northern Hemisphere, the Coriolis acceleration deflects trajectories to the right by several inches to feet depending on target latitude and azimuth.