Animating a wheel.

Hello!
First of all, because here we are talking about a spinning wheel, we should probably describe initial speed as degree per frame. So, say if the initial speed is x degree/frame, the the speed in n th frame should be S(n) = x - a(n^2). To make the wheel spin, you just need to call the correct function to rotate the wheel object S degree every frame. Here is some pseudo code(display() is the disply loop that should be called in every game loop ):

int frame = 0;
double Si = 0;
double S = 0;

display() {
//draw background
if (S > 0) {
S = Si - a*frame*frame;
frame++;
if (S > 0) {
rotate S degrees;
}
} else { //end of game, call what you should call }
swapBuffer();
}

Just my two sents, maybe -a(x^2) is not a good option, because it causes the wheel to slow down faster at the end. In a wheel game like this, usually the last few seconds are the most exciting ones, so maybe you should simply use S(n) = x- a*n instead.

int frame = 0;
double S = 0;

display() {
//draw background
if (S > 0) {
S -= a;
if (S > 0) {
rotate S degrees;
}
} else { //end of game, call what you should call }
swapBuffer();
}

to start the spin, simply assign a value(the initial speed) to S.

Hope your lucky wheel get you good luck!

That's a "bad path". Separate the physics from the graphics and make the graphics slave to the physics.
As-is this sounds like you are relying on the graphics calculations to achieve the spin-result.
This will result in non-deterministic behavior due to a variable calculation step-size (it'll be whatever the spurious free-running rate is... almost random).

Typically you pick a fixed rate to run the physics at. This makes the physics (much more) stable and repeatable.
If you update the graphics more frequently (don't have to) then you interpolate or predict & correct.

So write the physics part that takes an impulse and translates it into an initial rotational velocity and then tweak your spin-down.
Then make the rendering pass take the current angle from the physics engine and calculate a rotation matrix to apply to the wheel.

You create 'synthetic time' and update it at the physics step. Each step you run a pass of the physics engine.
When you have time (are caught up to real-time) you can render a frame. If you get all the work done you can sleep until real-time reaches the next physics step in synthetic-time.

*The primary problem is the inconsistent step size not rounding issues