Projectile Motion Calculator

It applies the standard kinematic equations under constant gravity g (default 9.80665 m/s^2): horizontal velocity vx = v0 cos(theta), vertical velocity vy(t) = v0 sin(theta) - g t, height y(t) = h0 + v0 sin(theta) t - 0.5 g t^2. Air resistance is ignored.

When launch height h0 is zero, the range simplifies to v0^2 sin(2 theta) / g. The sine of 2 theta peaks at 1 when 2 theta = 90 degrees, i.e. theta = 45 degrees. With a non-zero launch height, the optimal angle drops slightly below 45 degrees.

A positive launch height extends both. Flight time goes from 2 v0 sin(theta) / g to (v0 sin(theta) + sqrt((v0 sin(theta))^2 + 2 g h0)) / g, and the longer flight gives the horizontal velocity more time to carry the projectile, so range goes up too.

Air resistance depends on shape, surface roughness, spin, air density, and speed. Modeling it accurately requires a drag model and changes range and flight time from the no-drag result. This calculator uses the vacuum equations, the standard physics-class model.

Yes. Set gravity to 1.62 m/s^2 for the Moon, 3.71 for Mars, or 24.79 for Jupiter. Lower gravity stretches the range and flight time dramatically: the same 20 m/s launch at 45 degrees reaches about 41 m on Earth and 247 m on the Moon.