Projectile Impact Altitude Vacuum Calculator

Projectile trajectory in vacuum

Take away air resistance and gravity is the only thing left shaping the flight. The horizontal range comes out to R = v₀²·sin(2θ) / g, and the height at a given distance x is y(x) = x·tan(θ) − g·x² / (2·v₀²·cos²θ). On Earth g ≈ 9.81 m/s². On the Moon it drops to about 1.62 m/s², which is why the same shot carries roughly six times as far. There is a famous example: in 1971, Alan Shepard swung at a golf ball during Apollo 14, and with no atmosphere and feeble lunar gravity he claimed it went "miles and miles" (later analysis puts the second ball closer to 40 yards). In a real vacuum the range grows with v₀² and hangs entirely on the launch angle and gravity.

Applications

Planning lunar and Martian missions, from tossing rover samples to studying how a lander touches down. It also shows up in spacecraft ejecta and impact studies, in orbital mechanics classes, in vacuum-chamber experiments, and in railgun or coilgun ballistic models where the atmosphere barely matters.

FAQ

Why does the same shot fly farther on the Moon? Lunar gravity is about a sixth of Earth's, and there is no air to slow things down. Since range goes inversely with g, a 100 m shot here turns into roughly 600 m up there.

Is 45° still optimal in vacuum? Yes, and a vacuum is actually the only place where 45° really does win. Add an atmosphere and drag pulls the best angle down to about 30-40°, depending on what you are throwing.

Does altitude affect a vacuum calculation? Only by way of local gravity. On Mars, where g ≈ 3.71 m/s², the same projectile reaches about 2.6× farther than it would at sea level on Earth.