Position after a certain time

Newtonian Physics.

s = s₀ + v * t + ½a * t²

That is, to clear things up:

If your object starts at x = 0, has a xVelocity of xVel, and an acceleration of xAcc, then your position after T seconds (assuming units per second) is:

newX = 0 + T*xVel + 1/2 * a * T^2;

I know this is a repost of what Oluseyi said, but thought I'd explain the variables a little.

--Vic--

Indeed, now I remember the formula from some old mechanics classes. I had forgotten about it, such an essential one :D

But it doesn't totally work.

I do the following, every frame, for every particle:

    //update velocity    velX = velX + dts * accX;    velY = velY + dts * accY;    //update position    x = x + dts * velX + 0.5 * dts * dts * accX;    y = y + dts * velY + 0.5 * dts * dts * accY;

x and y are the position of the particle, velX and velY it's velocity, and accX and accY the force given to the particle by the emitter. dts is the time difference since the previous frame, in seconds.

Well I made an emitter that lets particles move around some gravity center, after running it for some time it looks like this:



Now, I pause the program for a second or so (by dragging the SDL window, because SDL then pauses the application in windows). I'd expect after this second, since the particles use the code I gave above, the particles would still be close to the center. dts will be something like 1 second then. However, they now do this:



For some reason they suddenly got much bigger velocities and manage to reach much farther.

Is there something incorrect about my code, like, the way to update velocity, or the order in which things happens?

EDIT:

Ok, I think I found why it happens: the accelaration isn't constant in time, it changes direction when the particles move around the gravity center.

How can I precalculate THAT? Only the direction of the accelaration changes, it's magnitude is constant.

For that, you'll need integral calculus. Yes, the magnitude of the acceleration is constant, but acceleration is a vector quantity (time to model it as one) and since the direction changes over an interval t...

Provided you can describe the change in a as a function of t, then you can do a simple integral:

integrate [s<sub>0</sub> + v * t + ½a * t² * da/dt][0, T]

where [0, T] is the interval you are integrating over. If you can't describe a as a function of t, then you're just going to have to run the simulation, though you can speed it up by not actually waiting for a full second, etc (just run it flat out, as fast as possible, disconnected from the game world timer).

The problem may be your velocity calculation. Oluseyi's equation needs a very small clarification. It should be

s = s₀ + v₀ * t + ½a * t²

It appears that in your code you update the velocity and then use the current velocity to get the position. You must use the original velocity (i.e. s₀ and v₀ are the position and velocity at t = 0). This is a VERY common mistake. The change to your code is simple: just calculate the position before updating the velocity.

    //update position    x = x + dts * velX + 0.5 * dts * dts * accX;    y = y + dts * velY + 0.5 * dts * dts * accY;   //update velocity    velX = velX + dts * accX;    velY = velY + dts * accY;