Projectile with air resistance

A point projectile moves under constant gravity and quadratic drag, F = −½ρ C_d A |v|v, through still air. Mass is positive; C_d is dimensionless and must use the same reference area A as its measurement. The default values are an illustrative ball example, not a calibrated baseball model. Density and C_d stay constant; wind, lift, spin, Earth curvature and changes with altitude or speed are omitted.

Calculate starts an adaptive RK4 calculation in a worker. Cancel, editing or navigation stops that job. Ground contact ends flight; the highest point is located within the simulated interval. If the chosen time horizon is reached first, the table shows observed distance/height/time and an unfinished state, not a claimed landing range. Open square endpoints mark unfinished curves. A finite step budget may reject very stiff combinations.

The solid line and its table column include drag; the dashed line is the analytic vacuum reference. Both use the same metre scale. Range difference is vacuum minus drag, expressed relative to the vacuum range only when both flights land and that range is positive. The extra quantities include β=ρ C_d A/(2m), terminal speed √(g/β), and integrated drag work per kg. With zero drag the analytic solution is reused.

Before Calculate, the plot shows only the analytic vacuum preview, not a drag simulation. Editing cancels the worker and restores this preview; Calculate explicitly compares both flights. Reset restores the default inputs. Exact input drafts, including incomplete or invalid entries, are kept locally; computed flights are never stored. Hover field labels/inputs for the full parameter names.