Brachistochrone (Flip-and-Burn) 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 burns continuously for the entire trip. Much faster, but it needs either a great deal of fuel or an engine far more efficient than anything flown. 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 ≈ 2.1 days. Peak velocity ≈ 877 km/s (0.29% of c).

The number that ends the daydream: delta-v. A ship burning at constant thrust spends velocity the whole way, and the total is simply a × t. For that Mars run it comes to about 1,754 km/s. A Hohmann transfer to Mars gets there on roughly 5.6 km/s, and the whole Apollo mission from pad to splashdown spent something like 18 km/s, so flip-and-burn is asking for about 300 Hohmann transfers or 100 Apollos in one trip. No chemical rocket gets near it. The trip is a fusion-drive or antimatter proposition, and that is exactly why flip-and-burn belongs to fiction rather than to any launch manifest.

For comfort, 0.3g is a gentle level of thrust, about like lying on a reclining seat. At 1g, occupants feel exactly as they would standing on Earth, which is what makes it the ideal cruise setting for a long voyage.