Modeling a projectile's motion

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So recently @Maximum_ADHD posted this on twitter and through a series of events I was asked if I would post a tutorial showing how to do it. I’m not sure if this is exactly how Clone did it, but this is how I approached it.

So to start off let’s cover the process that doing something like this would entail.

  1. Start with a position and velocity which we can use the above equations to find the projectile’s path.
  2. Find where this path intersects with in game geometry.
  3. Reflect the current velocity against the geometry’s surface.
  4. Use the intersection position and reflected velocity to repeat from step 1 as many times as desired.

We already know how to do steps 1 and 4 from the above post so all we need to cover are steps 2 and 3.

To find where our projectile intersects with a surface we’ll do two things.

We’ll first split our projectile’s path into multiple rays which can then be used to find an intersection with in game geometry.

local g = Vector3.new(0, game.Workspace.Gravity, 0)

local function x(t, v0, x0)
	return 0.5*g*t*t + v0*t + x0
end

local function intersection(x0, v0)
	local t = 0
	local hit, pos, normal
	
	repeat
		local p0 = x(t, v0, x0)
		local p1 = x(t + 0.1, v0, x0)
		t = t + 0.1
		
		local ray = Ray.new(p0, p1 - p0)
		hit, pos, normal = game.Workspace:FindPartOnRay(ray)
	until (hit or t > 5)
	
	-- so our plane is defined by the pos and normal variables
end

This will provide two key pieces of information; an estimated intersection position and a surface normal. We’ll then use those pieces of information to do a plane-quadratic intersection which will give us an exact intersection value.

Using a general form of the quadratic formula:

eq1

We know the quadratic will intersect with a plane defined by P and N when:

eq2

This gives us two situations where we solve for t differently.

The first being when a . N ≠ 0 in which case we use the quadratic formula.

eq3

The second being when a . N = 0 in which case we simple solve the linear equation for t:

eq4

In code this translates to:

local function planeQuadraticIntersection(v0, x0, p, n)
	local a = (0.5*g):Dot(n)
	local b = v0:Dot(n)
	local c = (x0 - p):Dot(n)
	
	if (a ~= 0) then
		local d = math.sqrt(b*b - 4*a*c)
		return (-b - d)/(2*a)
	else
		return -c / b
	end
end

All that’s left is to solve step 3 which is to reflect our velocity vector against the surface normal. This is simple enough and is currently explained on the dot product devhub wiki page. The only thing you might do here is slightly damp the velocity such that your reflection loses some energy as it bounces.

When you put all that together you get:

It’s not perfect mainly because how velocity is damped, but it’s a pretty good projection of the path traveled. Enjoy!

Trajectory.rbxl (25.2 KB)

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