I believe I might have spotted an error – well, more of an oversight in truth – with the Vector Movement rules.
The scale of the map cells, be them squares or hexes, was set to 648 Km as that is the distance an object accelerating at 10 m/s² will move over 360 seconds (a 6 minute turn). So far, so good: during the movement phase, you'll need to move your ship [Thrust] number of squares/hexes in the direction you've applied the thrust.
And then for next turn, the rules say, you've added that same value of [Thrust] to your velocity vector.
Therein lies the problem.
The Problem:
Consider the following; a ship with Thrust 1 has spent one turn (360 seconds) accelerating towards +X. What does this look like?
- How far has the ship travelled? From the equations of motion, we know that the distance travelled by the accelerated object is:
½ * (Acceleration) * (Time)². Doing the maths, we arrive at the aforementioned 648 Km result.
- What is the ship's velocity? This one is simpler: if the ship has accelerated at a rate of 10 m/s² for 360 seconds, at the end of those 360 seconds its speed will be
10 m/s² * 360 s = 3600 m/s.
So at the end of the movement phase, our ship will be travelling with a speed of 3600 m/s.
Let us then assume that the next turn, our ship does nothing but coast. Because it is not thrusting, we need not concern ourselves with the displacement that comes from acceleration, but we
do need to concern ourselves with the distance the ship will travel due to its current velocity.
How far
does it travel in a turn then, anyway? Simple: simply multiply its velocity by the length of a turn, and we'll know how far it'll travel:
3600 m/s * 360 s = 1296 Km. And there's the problem.
When a ship thrusts, it is displaced by one hex, yes, but its velocity vector doesn't increase by 1 hex/turn like the rules say, no, it increases by
2 hexes/turn.
The Solution:
Thankfully, the solution is really simple and just entails a rephrasing of the rules; Whenever the a ship applies thrust, it moves [Thrust] hexes in the direction of said Thrust, and its velocity increases by [Thrust * 2] in the direction of the thrust. That's really it, it works for any given value of thrust.
In case you want to know why, let us look at the maths for displacement through thrust and distance travelled by coasting side-by-side:
- Acceleration:
Distance = ½ * a * t²
- Coasting:
Distance = v * t
However, do notice that the velocity 'v' you inherit by undergoing a thrust of 'a' for one turn 't' can
also be expressed as
a * t.
So rewriting the above:
- Acceleration:
Distance = ½ * a * t²
- Coasting:
Distance = (a₁ * t₁) * t = a * t² (Obs.: where 'a₁' and 't₁' are the thrust and burn time values from the previous turn)
So the distance travelled due to those two is different only by a factor of exactly 2, hence why the rules' oversight can be fixed simply by saying the velocity vector increases by [Thrust * 2], not simply [Thrust].
