Draw the Shot First

The gold curve only works with one planet. Add a second mass and the formula breaks. The step doesn't care how many masses there are. So to predict a shot, we fly a ghost ball ahead of time with the same step.

The fixed step

Frames don't all take the same time. If the game stepped once per frame, a fast computer and a slow one would fly different paths. So the game never steps by the frame's time. It spends each frame's time in whole steps of 1/120 s, and carries what's left over into the next frame:

Three frames of 20 ms each above, and below them the 8.33 ms steps each frame pays for. Frames 1 and 2 buy two steps each, frame 3 buys three, and a sliver is left over for the next frame.

src/components/GravityBody/GravityBody.core.ts

/**
 * Spend `dt` seconds of frame time in whole fixed steps. Returns how many to
 * take and the time left over for next frame. The backlog beyond `maxSteps` is
 * DROPPED, not carried — carrying it would only make the next frame worse.
 */
export function planSteps(
  carried: number,
  dt: number,
  step: number = GRAVITY_STEP,
  maxSteps: number = MAX_STEPS_PER_FRAME,
): { steps: number; carry: number } {
  const total = carried + Math.max(0, dt);
  const steps = Math.floor(total / step);
  if (steps > maxSteps) return { steps: maxSteps, carry: 0 };
  return { steps, carry: total - steps * step };
}

planSteps(0, 1 / 60)  // { steps: 2, carry: 0 }
planSteps(0, 1 / 50)  // { steps: 2, carry: 0.003333 }   the leftover waits for the next frame
planSteps(0, 10)      // { steps: 12, carry: 0 }         a 10 s freeze takes 12 and drops the rest

So the balls take the same steps on a slow screen and a fast one. A ghost flown with the same step does the same sums, and lands in the same place.

Draw the Shot First

The fixed step

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Newton's Cannon: A Gravity Simulator in Arcade ECS

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