Any suggestions for Weapon Trail mathematics?

In this example, I will be using a ServerScript and the wait() function, however you will get nicer results if you bind it to a render-step.

If you want to make a trail for an object, you need to measure its displacement. Displacement is a vector quantity defined as the change in distance in a particular amount of time. Therefore, in our time loop, we want to keep track of what the previous position of the part was:

local part = game.Workspace.Part
while true do
    local prevPosition = part.Position
    wait()
    local displacement = part.Position - prevPosition
    trail(part, displacement)
end

Now we can create our trail. I will make it just a thin block that originates from the center of the object. I will separate the creation out into a function for readability:

function trail(object, displacement)
    -- Important aspects
    local trail = Instance.new("Part")
    trail.Size = Vector3.new(0, 0, displacement.magnitude)
    trail.CFrame = CFrame.new(object.Position - displacement/2, object.Position)

    trail.Parent = game.Workspace
    trail.Anchored = true
    trail.CanCollide = false
    game.Debris:AddItem(trail, 3)
end

Notice that for the size I used the magnitude of the displacement. This is because the distance of a vector is it's magnitude (distance = magnitude). Also note how I set the CFrame. That particular constuctor of CFrame takes two parameters, the position you want it to be centered and a position you want it to face. I center it at halfway between the previous position and the current position (current position - half displacement) and point it towards the current position.

One additional things to note about this example. You may want to make sure that there is a displacement. If the displacement of the object is very, very small (less than some epsilon) then you probably don't want a trail to be generated. This can be implemented like so:

if displacement.magnitude > 0.01 then
    trail(parameters)
else
    -- do nothing
end