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How to make verlet integration collisions more stable?

I'm not using any engine, but instead trying to build my own softbody dynamics for fun using verlet integeration. I made a cube defined by 4x4 points with segments keeping its shape like so:
I have the points collide against the edges of the scene and it seems to work fine. Though I do get some cases where the points collapses in itself and it'll create a dent instead of maintaining its box shape. For example, if it's a high enough velocity and it lands on its corner it tends to crumble:
I must be doing something wrong or out of order when solving the collision.
Here's how I'm handling it. It's in Javascript, though the language doesn't matter, feel free to reply with any language:
sim = function() {
// Sim all points.
for (let i = 0; i < this.points.length; i++) {
this.points[i].sim();
}
// Keep in bounds.
let border = 100;
for (let i = 0; i < this.points.length; i++) {
let p = this.points[i];
let vx = p.pos.x - p.oldPos.x;
let vy = p.pos.y - p.oldPos.y;
if (p.pos.y > height - border) {
// Bottom screen
p.pos.y = height - border;
p.oldPos.y = p.pos.y + vy;
} else if (p.pos.y < 0 + border) {
// Top screen
p.pos.y = 0 + border;
p.oldPos.y = p.pos.y + vy;
}
if (p.pos.x < 0 + border) {
// Left screen
p.pos.x = 0 + border;
p.oldPos.x = p.pos.x + vx;
} else if (p.pos.x > width - border) {
// Right screen
p.pos.x = width - border;
p.oldPos.x = p.pos.x + vx;
}
}
// Sim its segments.
let timesteps = 20;
for (let ts = 0; ts < timesteps; ts++) {
for (let i = 0; i < this.segments.length; i++) {
this.segments[i].sim();
}
}
}
Please let me know if I need to post any other details.

This may be better answered on a physics or game-dev exchange (and likely has already been), but i'll give it a crack because it's nice to revisit this stuff...
Verlet integration is a fantastically stable if not physically accurate method, but the problem here is not the integration method, or anything you've done wrong as far as I can see; it's the type of simulation: mass-aggregate physics (the building of geometry out of dynamic constraints), which is really nice and simple :) but has some inherent deficiencies and limitations, and this particular problem is inherent to the simulation type.
First, look carefully at the arrangement of constraints in the collapsed box - they are just as valid as the initial one. Although individual constraints may not be as satisfied, in combination they are still in a local equilibrium with each other - there nothing compelling them to form their original arrangement.
Second, the external force (collision with immovable plane) is what overcame the constraint forces originally. Even if the constraints respond proportionally towards infinity as they compress - the simulation can never match reality because it works a frame at a time not continuously, and the longer the frame the more error.
To retain shape more reliably, a more explicit angular constraint is needed, which usually involves quaternions - these are quite a bit harder to implement than distance constraints, and once you have them you will be pretty far along the road to implementing rigid body physics anyway. But there are ways to mitigate instead:
1. Use a smaller interval
All posteriori simulations and numerical integration in general has some inherent instability. While different integration methods (e.g verlet) can mitigate this, generally the smaller the interval the better the stability. This alone will give the constraints more opportunity to push back against external forces, but it will also increase the maximum stable constraint stiffness.
This will probably require you to optimise your engine more. Additionally make sure you don't couple your render step to the simulation, you want to be able to render at multiples of the simulation interval which allows you to run the simulation faster for stability without unnecessarily rendering more frames than is useful, as it will just slow down the simulation.
2. Try more stable shapes
For your box-of-boxes shape, see what happens when you add more constraints between far vertices, it will add more global stability to the shape.
A common type of shape people make with mass-aggregate physics are polygons, because it's straight forward to make them highly interconnected (a little bit like a bicycle wheel, but where each spoke point connects to every other spoke point). In- fact spoke-like designs are one of the most stable, but you can usually apply the same principles to more irregular shapes once you get an intuition for it.
3. A different type of constraint
Quaternions are not the only possible constraint that will help retain configuration, the main problem is not so much that explicit angle isn't being maintained; but rather that when points are forced past a certain position relative to their siblings, their distance constraints flip and start working in reverse - keeping them on that side.
There could be many different ways to solve this without something as complex as quarternions - in fact I will give it some thought and edit this post if I come up with anything, I have a bit more ammunition since I last explored mass-aggregate physics...

Acceleration over time

Scenario
I am working on a top-down view game in which enemies move towards certain positions. The destination being moved towards will often change drastically - sometimes while the enemy is still in motion towards the previous destination...
I want to achieve movement which is more realistic than linear movement, so there should be some acceleration and deceleration as enemies switch between the destinations.
Steering (direction) is not a factor. You may assume the sprites will move much like a hovercraft, drifting between the destinations as quickly as it can manage with respect to acceleration and deceleration.
For simplicity - lets assume there is only 1 dimension (Y) instead of X and Y... the movement should emulate a car which can only move north or south.
Since we are considering realistic movement in this scenario, you might not be surprised that a maximum speed over time is also a consideration. The enemy should never exceed it's own maximum speed; enemies store their own maximum speed in a variable.
One final consideration is that enemies will not only have a 'maximum speed' value, but it will also have a 'maximum acceleration' value - this would be the thing which guides how quickly each enemy can respond to moving in the opposite direction.
For simplicity, assume that the enemy does not have any movement friction... when it stops accelerating, it just keeps cruising at the current velocity forever.
Example
For context, lets elaborate on the car example. A particular car has:
Maximum speed: 10 meters per second
Maximum acceleration: can reach top speed in 2 seconds
(Other factors like destination y-pos, position y-pos, current velocity, etc)
Just like when I'm driving a car, I imagine all of these values are present, but I can't change them. All I can really change is how much I'm putting my foot on the acceleration/brake. Let's call this the 'throttle'. Like the acceleration pedal in a car, I can change this value to any amount at any time, in no time at all.
I can plant my foot down (throttle=1), let go of the pedal immediately (throttle=0), and even change into reverse and plant my foot down again (throttle=-1). Lets assume these changes in throttle are INSTANT (unlike speed or acceleration, which grow/shrink over TIME)
All that said, I imagine the only value that I really need to calculate is what the throttle should be, since thats the only thing I can control in the vehicle.
So, how then do I know how much throttle to use at any given moment, to arrive as quickly as possible to a destination (like some traffic lights) without overshooting my destination? I will need to know how much to push down the accelerator, briefly not accelerate at all, and then how hard decelerate as I'm closing in on the destination.
Preempted movement
This game will likely have an online component. That said, players will be transmitting their positions over a socket connection... however even the best connection could never achieve sending positions frequently enough to achieve smooth movement - you need interpolation. You need to send the 'velocity' (speed) along with the co-ordinates so that I can assume future positions in the time between packets being received.
For that reason, Tweens are a no-go. There would be no way to send a tween, then accurately inform the other parties at what point along each tween each entity currently is (I mean, I guess it is possible, but horrendously overcomplicated and probably involves rather large packet sends, and is probably also very exploitable in terms of the online component), then you need to consider aborting Tweens as the destinations change, etc.
Don't get me wrong, I could probably already model some very realistic movement with the ease-in/ease-out functionality of Tweens, and it would look great, but in an online setting that would be.. very messy.
How's it look so far?
So, essentially I have established that the primary aspect which needs to be calculated at any time is how much throttle to use. Let's work through my logic...
Imagine a very basic, linear movement over time formula... it would look like this:
Example 1 - position over time
currentDY = 5; // Current 'velocity' (also called Delta Y or 'DY')
currentY += currentDY * time // Current Y pos is increased by movement speed over time.
As you can see, at any given moment, the Y position increases over time due to the 'velocity' or DY (over time). Time is only a factor ONCE, so once we reach the destination we just set the DY to zero. Very sharp unrealistic movement. To smoothen the movement, the velocity ALSO needs to change over time...
Example 2 - velocity over time
throttle = -1
currentDY += throttle * time;
currentY += (currentDY * time);
//Throttle being -1 eventually decelerates the DY over time...
Here, the throttle is '-1' (maximum reverse!), so over time this will reduce the velocity. This works great for realistic acceleration, but provides no realistic anticipation or deceleration.
If we reach the destination this time, we can set the throttle to '0'... but it's too late to brake, so the resulting enemy will just keep moving past the target forever. We could throttle = '1' to go back, but we'll end up swinging back and forth forever.
(Also note that the maximum acceleration and speed isn't even a factor yet - it definately needs to be! The enemy cannot keep ramping up their speed forever; velocity delta AND ALSO acceleration delta need to have limits).
All that said, it's not enough to simply change velocity over time, but we also need be able to anticipate how much to decelerate (i.e. 'backwards throttle') BEFORE IT HAPPENS. Here's what I've got so far, and I'm practically certain this is the wrong approach...
Example 3 - throttle over time?? (I'm stuck)
guideY = currentY + (currentDY * (timeScale * 3000));
dist = endY - guideY;
throttle = Math.max(-1, Math.min(1, dist / 200));
currentDY += throttle * time;
currentY += (currentDY * time);
As you can see, this time we attempt to anticipate how much throttle to use by guessing where the enemies position will be in an arbitrary time in the future (i.e. 3 seconds). If the guideY went past the destination, we know that we have to start BRAKING (i.e. reducing speed to stop on top of the destination). By how much - depends on how far away the enemies future position is (i.e. throttle = dist / 200;)
Here's is where I gave up. Testing this code and changing the values to see if they scale properly, the enemy always swings way over the destination, or takes far too long to 'close in' on the destination.
I feel like this is the wrong approach, and I need something more accurate. It feels like I need an intersection to correctly anticipate future positions.
Am I simply using the wrong values for the 3 seconds and dist / 200, or am I not implementing a fully working solution here?
Presently, compared to the linear movement, it always seems to take 8 times longer to arrive at the target position. I haven't even reached the point of implementing maximum values for DeltaVelocity or DeltaAcceleration yet - the accepted solution MUST consider these values despite not being present in my JSFiddle below...
Test my logic
I've put all my examples in a working JSFiddle.
JSFiddle working testbed
(Click the 'ms' buttons below the canvas to simulate the passing of time. Click a button then press+hold Return for very fast repetition)
The sprite is initially moving in the 'wrong' direction - this is intended for testing robustness - It assumes an imaginary scenario where we just finished moving as fast as possible toward an old destination far lower on the screen, and now we need to suddenly start moving up...
As you can see, my third example (see the update function), the time for the sprite to 'settle' at the desired location takes far longer than it should. My math is wrong. I can't get my head around what is needed here.
What should the throttle be at any given time? Is using throttle even a correct approach? Your assistance is very appreciated.
Tiebreaker
Alright, I've been stuck on this for days. This question is going up for some phat bounty.
If a tiebreaker is required, the winner will need to prove the math is legit enough to be tested in reverse. Here's why:
Since the game also comprises of a multiplayer component, enemies will be transmitting their positions and velocities.
As hacking protection, I will eventually need a way to remotely 'check' if the velocity and position at between any two sample times is possible
If the movement was too fast based on maximum velocity and acceleration, the account will be investigated etc. You may assume that the game will know the true maximum acceleration and velocity values of the enemies ahead of time.
So, as well as bounty, you can also take satisfaction in knowing your answer will contribute to ruining the lives of filthy video game cheaters!

Edit 2: Fiddle added by answer author; adding into his answer in case anyone else finds this question: http://jsfiddle.net/Inexorably/cstxLjqf/. Usage/math explained further below.
Edit 1: Rewritten for clarification received in comments.
You should really change your implementation style.
Lets say that we have the following variables: currentX, currentY, currentVX, currentVY, currentAX, currentAY, jerk.
currentVX is what would have been your currentDX. Likewise, currentAX is the x component of your delta velocity value (accel is derivative of velocity).
Now, following your style, we're going to have a guideX and a guideY. However, there is another problem with how you're doing this: You are finding guideY by predicting the target's position in three seconds. While this is a good idea, you're using three seconds no matter how close you are to the target (no matter how small dist is). So when the sprite is 0.5 seconds from the target, it's going to still be moving towards the target's estimated position (three seconds into the future). This means it won't actually be able to hit the target, which seems to be the problem that you implied.
Moving on, recall the previous variables that I mentioned. Those are the current variables -- ie, they will be updated at every call after some seconds have been passed (like you have been doing before). You also mentioned a desire to have maxV, and maxA.
Note that if you have a currentVX of 5 and a currentVY of 7, the velocity is (5^2+7^2)^0.5. So, what you're going to want to do each time you're updating the 'current' archetype of variables is before updating the value, see if the magnitude (so sqrt(x^2+y^2) of those variables like I showed with velocity) will exceed the respective maxV, maxA, or jmax values that you have set as constants.
I'd also like to improve how you generate your guide values. I'm going to assume that the guide can be moving. In this case, the target will have the values listed above: x, y, vx, vy, ax, ay, jx, jy. You can name these however you'd like, I'm going to use targetX, targetY... etc to better illustrate my point.
From here you should be finding your guide values. While the sprite is more than three seconds away from the target, you may use the target's position in three seconds (note: I recommend setting this as a variable so it is simple to modify). For this case:
predictionTime = 3000*timescale; //You can set this to however many seconds you want to be predicting the position from.
If you really wanted to, you could smooth the curve using integration functions or loops for the guide values to get a more accurate guide value from the target values. However, this is not a good idea because if you ever implement multiple targets / etc, it could have a negative impact on performance. Thus, we will use a very simple estimation that is pretty accurate for being such low cost.
if (sprite is more than predictionTime seconds away from the target){
guideX = targetX + predictionTime * targetVX;
guideY = targetY + predictionTime * targetVY;
}
Note that we didn't account for the acceleration and jerk of the target in this, it's not needed for such a simple approximation.
However, what if the sprite is lese than predictionTime seconds away from the target? In this case we want to start increasingly lessening our predictionTime values.
else{
guideX = targetX + remainingTime * targetVX;
guideY = targetY + remainingTime * targetVY;.
}
Here you have three choices on finding the value of remaining time. You can set remainingTime to zero, making the guide coordinates the same as the targets. You can set remainingTime to be sqrt((targetX-currentX)^2+(targetY-currentY))/(sqrt(currentVX)^2+(currentVY)^2), which is basically distance / time in 2d, a cheap and decent approximation. Or you can use for loops as mentioned before to simulate an integral to account for the changing velocity values (if they are deviating from maxV). However, generally you will be remaining at or close to maxV, so this is not really worth the expense. Edit: Also, I'd also recommend setting remainingTime to be 0 if it is less than some value (perhaps about 0.5 or so. This is because you don't want hitbox issues because your sprite is following guide coordinates that have a slight offset (as well as the target moving in circles would give it a larger velocity value while changing direction / essentially a strong evasion tactic. Maybe you should add something in specifically for that.
We now have the guideX and guideY values, and account for getting very close to a moving target having an effect on how far from the target the guide coordinates should be placed. We will now do the 'current' value archetype.
We will update the lowest derivative values first, check to see if they are within bounds of our maximum values, and then update the next lowest and etc. Note that JX and JY are as mentioned before to allow for non constant acceleration.
//You will have to choose the jerk factor -- the rate at which acceleration changes with respect to time.
//We need to figure out what direction we're going in first. Note that the arc tangent function could be atan or whatever your library uses.
dir = arctan((guideY-currentY)/(guideX-currentX));
This will return the direction as an angle, either in radians or degree depending on your trig library. This is the angle that your sprite needs to take to go in the direction of guide.
t = time; //For ease of writing.
newAX = currentAX + jerk*t*cos(dir);
newAY = currentAY + jerk*t*sin(dir);
You may be wondering how the newAx value will ever decrease. If so, note that cos(dir) will return negative if guide is to the left of the target, and likewise sin(dir) will return negative if the sprite needs to go down. Thus, also note that if the guide is directly below the sprite, then newAx will be 0 and newAY will be a negative value because it's going down, but the magnitude of acceleration, in other words what you compare to maxA, will be positive -- as even if the sprite is moving downwards, it's not moving at negative speed.
Note that because cos and sin are of the same library as atan presumably, so the units will be the same (all degrees or all radians). We have a maximum acceleration value. So, we will check to make sure we haven't exceeded that.
magnitudeA = sqrt(newAX^2+newAY^2);
if (magnitudeA > maxA){
currentAX = maxA * cos(dir);
currentAY = maxA * sin(dir);
}
So at this point, we have either capped our acceleration or have satisfactory acceleration components with a magnitude less than maxA. Let us do the same for velocity.
newVX = currentVX + currentAX*t;
newVY = currentVY + magnitudeA*t*sin(dir);
Note that I have included two ways to find the velocity components here. Either one works, I'd recommend choosing one and using it for both x and y velocity values for simplicity. I just wanted to highlight the concept of magnitude of acceleration.
magnitudeV = sqrt(newVX^2+newVY^2);
if (magnitudeV > maxV){
currentVX = maxV * cos(dir);
currentVY = maxV * sin(dir);
}
We'd also like to stop boomeranging around our target. However, we don't want to just say slow down alot like you did in your JSFiddle, because then if the target is moving it will get away (lol). Thus, I suggest checking how close you are, and if you are in a certain proximity, reduce your speed linearly with distance with an offset of the target's speed. So set closeTime to something small like 0.3 or what ever feels good in your game.
if (remainingTime < closeTime){
//We are very close to the target and will stop the boomerang effect. Note we add the target velocity so we don't stall while it's moving.
//Edit: We should have a min speed for the enemy so that it doesn't slow down too much as it gets close. Lets call this min speed, relative to the target.
currentVX = targetVX + currentVX * (closeTime - remainingTime);
currentVY = targetVY + currentVY * (closeTime - remainingTime);
if (minSpeed > sqrt(currentVX^2+currentVY^2) - sqqrt(targetVX^2-targetVY^2)){
currentVX = minSpeed * cos(dir);
currentVY = minSpeed * sin(dir);
}
}
magnitudeV = sqrt(currentVX^2+currentVY^2);
At this point we have good values for velocity too. If you going to put in a speedometer or check the speed, you're interested in magnitudeV.
Now we do the same for position. Note you should include checks that the position is good.
newX = currentX + currentVX*t;
newY = currentY + currentVY*t;
//Check if x, y values are good.
current X = newX; currentY = newY;
Now everything has been updated with the good values, and you may write to the screen.

Maths and Easing For Javascript/Canvas Preload Animation

I'm building an image preload animation, that is a circle/pie that gets drawn. Each 'slice' is totalImages / imagesLoaded. So if there are four images and 2 have loaded, it should draw to 180 over time.
I'm using requestAnimFrame, which is working great, and I've got a deltaTime setup to restrict animation to time, however I'm having trouble getting my head around the maths. The closest I can get is that it animates and eases to near where it's supposed to be, but then the value increments become smaller and smaller. Essentially it will never reach the completed value. (90 degrees, if one image has loaded, as an example).
var totalImages = 4;
var imagesLoaded = 1;
var currentArc = 0;
function drawPie(){
var newArc = 360 / totalImages * this.imagesLoaded // Get the value in degrees that we should animate to
var percentage = (isNaN(currentArc / newArc) || !isFinite(currentArc / newArc)) || (currentArc / newArc) > 1 ? 1 : (currentArc / newArc); // Dividing these two numbers sometimes returned NaN or Infinity/-Infinity, so this is a fallback
var easeValue = easeInOutExpo(percentage, 0, newArc, 1);
//This animates continuously (Obviously), because it's just constantly adding to itself:
currentArc += easedValue * this.time.delta;
OR
//This never reaches the full amount, as increments get infinitely smaller
currentArc += (newArc - easedValue) * this.time.delta;
}
function easeInOutExpo(t, b, c, d){
return c*((t=t/d-1)*t*t + 1) + b;
}
I feel like I've got all the right elements and values. I'm just putting them together incorrectly.
Any and all help appreciated.

You've got the idea of easing. The reality is that at some point you cap the value.
If you're up for a little learning, you can brush up on Zeno's paradoxes (the appropriate one here being Achilles and the Tortoise) -- it's really short... The Dichotomy Paradox is the other side of the same coin.
Basically, you're only ever half-way there, regardless of where "here" or "there" may be, and thus you can never take a "final"-step.
And when dealing with fractions, such as with easing, that's very true. You can always get smaller.
So the solution is just to clamp it. Set a minimum amount that you can move in an update (2px or 1px, or 0.5px... or play around).
The alternative (which ends up being similar, but perhaps a bit jumpier), is to set a threshold distance. To say "As soon as it's within 4px of home, snap to the end", or switch to a linear model, rather than continuing the easing.

Vectors, calculate movement forces with max speed

Im building a small space shooter game. I have how ever stubbed on a math problem when it come to the space physics.
Describing this with words is following:
There is a max speed.
So if you give full full speed you ship will move with this over the screen over and over again like in the old asteroids games.
If then release the rocket boost you ship should be keep moving with that speed over the screen.
Then the tricky part where Im stuck right now.
If you rotate the ship ANY angle and gives boost again the ship should try to get to this direction and NEVER surpas the max speed when it comes to how fast it is moving. so my question is. anyone have a good idea formula for this issue? feels like it has been done before if you know what to look for. :)
Ill add this small image to illustrate what is tried to be done with some vector calculations.
Red ring: Max speed
Green line: current ship direction.
Black line: direction(s) and how fast the ship is moveing in x and y.
Black ring: origin of movement.
Can illustrate it but hard to find a good math solution for this. :)
EDIT
This is the code Im using right now in every frame. It gives movement to the ship but does not give the force of movement the user has to counter-react with its rocket boosters to get the ship to stop or slow down. With this it stops the instant you release the accelerating speed for the ship.
//Calculates ship movement rules
var shipVelocityVec = GetVectorPosByAngle(shipMoveSpeed, shipRotationAngle);
var shipUnitVec =$V([Math.cos(shipRotationAngle),Math.sin(shipRotationAngle),0]);
var rawAccel = shipAccelSpeed / shipMass;
var scale = (shipUnitVec.dot(shipVelocityVec))/(shipTopSpeed * shipTopSpeed);
var v1 = shipUnitVec.subtract(shipVelocityVec.multiply(scale));
var finalAccelVec = v1.multiply(rawAccel);
console.log(finalAccelVec);
//move ship according to rules
var shipPosVector = $V([shipxPos, shipyPos, 0]);
var movementVector = shipPosVector.add(finalAccelVec);
shipxPos = movementVector.elements[0];
shipyPos = movementVector.elements[1];
To give the acceleration speed the user has to keep the button pressed. the instance the user releases the button the acceleration is set to zero and have to boost over again to give maximum acceleration throttle.
Solution found! Posted it here how it was done.

You seem to be confusing something - there is no issue capping the velocity to a maximum speed, even when the acceleration is at a different angle, if you are using vectors correctly.
Your setup should look something like this:
Your ship should have a position, a velocity, and an acceleration. Each of these can be represented as a 2D vector (with separate x and y components).
Every frame, add the velocity to the position, and the acceleration to the velocity.
Every frame, check that the speed does not exceed some maximum. If it does, cap the speed by normalizing the velocity vector and multiplying it by the max speed.
That's it! There are no special cases to consider - that's the magic of vector algebra!

#BlueRaja's solution should work, although you will get an abrupt change in behavior when you hit the max speed.
If you want a solution with no "seams", I believe you can do what you're looking for by adding the right kind of adjustment to the acceleration, as follows:
ship_unit_vec = [cos(ship_angle), sin(ship_angle)]
raw_accel = (engine_thrust / ship_mass)
scale = dot_product(ship_unit_vec, ship_velocity_vec) / max_speed^2
final_accel_vec = raw_accel * (ship_unit_vec - scale * ship_velocity_vec)
Notes:
If |ship_velocity_vec|<<max_speed, the scale * ship_velocity_vec component is negligable.
If |ship_velocity_vec|==max_speed, the scale * ship_velocity_vec component cancels all additional acceleration in the "wrong" direction, and aids acceleration in the "right" direction.
I've never tried this out, so I don't know how it will feel to the player...
More generally, if there are more sources of acceleration than just the ship thrusters, you can add them all together (say, as raw_accel_vec), and perform the above operation all at once:
scale_forall = dot_product(raw_accel_vec, ship_velocity_vec) / max_speed^2
final_accel_vec = raw_accel_vec - scale_forall * ship_velocity_vec

Rather than just imposing an ad hoc maximum speed, you could use some actual physics and impose a drag force. This would be an extra force acting on the spaceship, directed opposite to the velocity vector. For the magnitude of the drag force, it's simplest to just take it proportional to the velocity vector.
The overall effect is that the drag force increases as the spaceship moves faster, making it harder to accelerate in the direction of motion when the ship moves faster. It also makes acceleration easier when it is opposed to the direction of motion.
One point where this diverges from your description is that the spaceship won't continue at maximum speed forever, it will slow down. It won't, however, come to a halt, since the drag force drops as the ship slows down. That matches my memory of asteroids better than the ship continuing forever at constant velocity, but it has been quite a while since I've played.

I did it! Thank you for your help.
Finally found the solution. the problem was I was trying to modify the ships current movement when it comes to speed and then of this calculate the "drag" forces that would be the product of this movement when the user tried to go another direction. The solution was like #BlueRaja and #Comingstorm mentioned. All forces should be added together when it comes to the movement. This should be what then alter the ships position. Should not be added to the ships current movement. You might be able to effect a current movement to but then you have to do this differently. So I thought I share my solution for this how it looks.
This function is run each time the user accelerates the ship.
function CalcShipMovement() {
//Calculates ship movement rules
shipPosVector = $V([shipxPos, shipyPos, 0]);
var shipVelocityVec = GetVectorPosByAngle(shipAccelSpeed, shipRotationAngle);
var shipUnitVec = $V([Math.cos(shipRotationAngle), Math.sin(shipRotationAngle), 0]);
if(currentShipMoveVector != null && Get2DVectorLength(currentShipMoveVector) > 0) {
var nextMove = currentShipMoveVector.add(shipVelocityVec);
var nextSpeed = Get2DVectorLength(nextMove);
//check if topspeed of movement should be changed
if(nextSpeed > shipTopSpeed) {
var scale = nextSpeed / shipTopSpeed;
currentShipMoveVector = DevideVector(nextSpeed, scale);
} else {
currentShipMoveVector = currentShipMoveVector.add(shipVelocityVec);
}
}
if(currentShipMoveVector != null && Get2DVectorLength(currentShipMoveVector) == 0) {
currentShipMoveVector = currentShipMoveVector.add(shipVelocityVec);
}}
This code is run in every frame the graphics for the ship is generated to alter its position.
function SetShipMovement() {
if(currentShipMoveVector != null && Get2DVectorLength(currentShipMoveVector) > 0) {
shipMoveSpeed = Get2DVectorLength(currentShipMoveVector);
shipPosVector = shipPosVector.add(currentShipMoveVector);
shipxPos = shipPosVector.elements[0];
shipyPos = shipPosVector.elements[1];
//Makes the ship slow down if no acceleration is done for the ship
if(shipAccelSpeed == 0) {
currentShipMoveVector = currentShipMoveVector.subtract(DevideVector(currentShipMoveVector, 50));
}
} else {
currentShipMoveVector = $V([0, 0, 0]);
}}

2d parabolic projectile

I'm looking to create a basic Javascript implementation of a projectile that follows a parabolic arc (or something close to one) to arrive at a specific point. I'm not particularly well versed when it comes to complex mathematics and have spent days reading material on the problem. Unfortunately, seeing mathematical solutions is fairly useless to me. I'm ideally looking for pseudo code (or even existing example code) to try to get my head around it. Everything I find seems to only offer partial solutions to the problem.
In practical terms, I'm looking to simulate the flight of an arrow from one location (the location of the bow) to another. I have already simulated the effects of gravity on my projectile by updating its velocity at each logic interval. What I'm now looking to figure out is exactly how I figure out the correct trajectory/angle to fire my arrow at in order to reach my target in the shortest time possible.
Any help would be greatly appreciated.

Pointy's answer is a good summary of how to simulate the movement of an object given an initial trajectory (where a trajectory is considered to be a direction, and a speed, or in combination a vector).
However you've said in the question (if I've read you correctly) that you want to determine the initial trajectory knowing only the point of origin O and the intended point of target P.
The bad news is that in practise for any particular P there's an infinite number of parabolic trajectories that will get you there from O. The angle and speed are interdependent.
If we translate everything so that O is at the origin (i.e. [0, 0]) then:
T_x = P_x - O_x // the X distance to travel
T_y = P_y - O_y // the Y distance to travel
s_x = speed * cos(angle) // the X speed
s_y = speed * sin(angle) // the Y speed
Then the position (x, y) at any point in time (t) is:
x = s_x * t
y = s_y * t - 0.5 * g * (t ^ 2)
so at impact you've got
T_x = s_x * t
T_y = -0.5 * g * (t ^ 2) + s_y * t
but you have three unknowns (t, s_x and s_y) and two simultaneous equations. If you fix one of those, that should be sufficient to solve the equations.
FWIW, fixing s_x or s_y is equivalent to fixing either speed or angle, that bit is just simple trigonometry.
Some combinations are of course impossible - if the speed is too low or the angle too high the projectile will hit the ground before reaching the target.
NB: this assumes that position is evaluated continuously. It doesn't quite match what happens when time passes in discrete increments, per Pointy's answer and your own description of how you're simulating motion. If you recalculate the position sufficiently frequently (i.e. 10s of times per second) it should be sufficiently accurate, though.

I'm not a physicist so all I can do is tell you an approach based on really simple process.
Your "arrow" has an "x" and "y" coordinate, and "vx" and "vy" velocities. The initial position of the arrow is the initial "x" and "y". The initial "vx" is the horizontal speed of the arrow, and the initial "vy" is the vertical speed (well velocity really but those are just words). The values of those two, conceptually, depend on the angle your bowman will use when shooting the arrow off.
You're going to be simulating the progression of time with discrete computations at discrete time intervals. You don't have to worry about the equations for "smooth" trajectory arcs. Thus, you'll run a timer and compute updated positions every 100 milliseconds (or whatever interval you want).
At each time interval, you're going to add "vx" to "x" and "vy" to "y". (Thus, note that the initial choice of "vx" and "vy" is bound up with your choice of time interval.) You'll also update "vx" and "vy" to reflect the effect of gravity and (if you feel like it) wind. If "vx" doesn't change, you're basically simulating shooting an arrow on the moon :-) But "vy" will change because of gravity. That change should be a constant amount subtracted on each time interval. Call that "delta vy", and you'll have to tinker with things to get the values right based on the effect you want. (Math-wise, "vy" is like the "y" component of the first derivative, and the "delta vy" value is the second derivative.)
Because you're adding a small amount to "vy" every time, the incremental change will add up, correctly simulating "gravity's rainbow" as your arrow moves across the screen.
Now a nuance you'll need to work out is the sign of "vy". The initial sign of "vy" should be the opposite of "delta vy". Which should be positive and which should be negative depends on how the coordinate grid relates to the screen.
edit — See #Alnitak's answer for something actually germane to your question.