Projectile Motion Calculator

Projectile motion is what happens to anything launched into the air and left to gravity alone: a thrown ball, a kicked shot, a fired shell. The whole problem simplifies the moment the launch velocity v at angle θ is split into two independent components — a horizontal speed vx = v·cos(θ) that never changes during the flight, and a vertical speed vy = v·sin(θ) that gravity steadily erases and then reverses. This calculator performs exactly that decomposition first and shows the numbers plugged in, so vx and vy are not hidden constants but a visible, checkable step.

From vx and vy the rest follows by substitution. On flat ground the time of flight is t = 2·vy/g (the time for gravity to cancel and then rebuild the initial vertical speed), the peak height is h = vy²/(2g), and the range is R = vx·t, which is the same as the textbook single-line formula R = v²·sin(2θ)/g. When the launch point sits above the landing point — a ball thrown off a cliff, a shot put released above shoulder height — the flat-ground shortcut for time no longer applies, so the calculator instead solves the full quadratic 0 = h0 + vy·t − ½·g·t² with the quadratic formula and keeps the positive root, then reuses that exact time of flight for the range.

The trajectory itself is drawn, not implied: the calculator samples roughly eighty points of x(t) = vx·t and y(t) = h0 + vy·t − ½·g·t² across the real computed flight time and renders them as a scaled SVG path, with gridlines, numeric axis ticks, the ground line, and dedicated markers for the highest point of the arc and where it lands. Because the chart comes from the same numbers as the derivation above it, changing the speed, angle or launch height changes the shape of the curve exactly the way the physics predicts.

The gravitational constant is stated explicitly with every result — 9.8 m/s² in the metric unit system, 32.2 ft/s² in the imperial one — since a projectile problem is meaningless without knowing which value of g was used. For a different flavor of stepped-out algebra, the quadratic equation solver on this site shows the same quadratic-formula substitution used here for elevated launches applied to any ax² + bx + c = 0, and the triangle calculator handles trigonometry problems that are not about motion at all. Everything runs locally in your browser: nothing is uploaded, there is no registration, and you can rerun as many launch angles as you like.