Brachistochrone Trajectory Calculator

The brachistochrone trajectory (also called the “flip-and-burn” maneuver) is the fastest practical path between two planets using constant thrust. The ship accelerates at full power for the first half of the journey, then flips 180° and decelerates at the same rate to arrive at the destination with near-zero velocity. This is far faster than a conventional Hohmann transfer orbit, which coasts unpowered for months or years.

Why Flip-and-Burn? A Hohmann transfer uses the minimum fuel but takes the longest time. A brachistochrone trajectory uses continuous thrust for the entire trip — much faster, but requires a great deal of fuel or a very efficient engine. Science fiction ships capable of 1g constant acceleration could reach Mars in a matter of days instead of months.

The Formula (non-relativistic): For total distance d and constant acceleration a:

t = 2 × √(d / a)

This comes from the kinematics of constant acceleration: d = ½at², so for one leg (half the trip): d/2 = ½a(t/2)², giving t = 2√(d/a).

Peak Velocity (at the midpoint, where flip occurs): v_max = a × (t / 2)

This is the maximum speed reached at the halfway point, just before the flip.

Planetary Distances: Planets move in elliptical orbits, so the distance between them changes constantly. This calculator uses the simplified model of average orbital distances from the Sun (in AU). The actual distance at any given time depends on where each planet is in its orbit.

Example — Earth to Mars at 1g: Distance ≈ 0.524 AU ≈ 78.4 million km. Acceleration = 9.80665 m/s². Total time ≈ 2.5 days. Peak velocity ≈ 1,060 km/s (0.35% of c).

For comfort, 0.3g is a gentle level of thrust (like lying on a reclining seat). At 1g, occupants feel exactly like standing on Earth — ideal for long voyages.