Projectile motion calculator

Projectile motion is two independent one-dimensional problems that share a clock. Horizontally nothing accelerates, so the velocity is constant. Vertically gravity pulls down at a constant rate. Once you separate them, everything else falls out of the kinematic equations.

The four results people usually want

Time of flight. Set y = 0 and solve the quadratic. Launching from ground level it simplifies to t = 2 v₀ sin(θ) / g. From a height it does not, which is why the calculator above solves the full form.

Maximum height. The vertical velocity is zero at the apex, so h_max = h₀ + (v₀ sin θ)² / (2g).

Range. Multiply the horizontal velocity by the total time. From ground level this gives the familiar R = v₀² sin(2θ) / g.

Impact velocity. The horizontal component never changed. The vertical one is v_y − g t. Combine them with Pythagoras.

Why 45 degrees is the optimum, and when it is not

From flat ground and ignoring air, the range term sin(2θ) peaks when 2θ = 90°, so θ = 45°. Two useful consequences: complementary angles give the same range — 30° and 60° land in the same place — and the shallow one arrives sooner and flatter.

Launch from a height and the symmetry breaks. The projectile spends extra time falling past the launch level, and that extra time is worth more if it is being spent moving horizontally. The optimum drops below 45°, approaching zero as the launch height dominates. Throwing a ball off a cliff, you want a flat throw.

What this model leaves out

• Air resistance. Drag scales roughly with the square of speed. For a golf ball or a bullet the real range is a fraction of the vacuum prediction, and the trajectory becomes asymmetric — steeper on the way down than on the way up.
• Lift and spin. A spinning ball generates sideways force through the Magnus effect, which is the entire basis of a curveball.
• Variation in g. Constant gravity is fine over a few kilometres and wrong for ballistic missiles or orbits.
• Earth’s rotation. The Coriolis deflection is negligible for a thrown object and significant for long-range artillery.

Gravity elsewhere