You'd be good to do some reading up on derivatives/integrals and how they map to position, velocity, and acceleration.

Position is what you think. Velocity, is change in position for a given time. So if you're moving 10 feet, and your logic loop is 1 second, velocity is 10 feet per second.

So velocity is delta V / T. (Change in V per time)

And to use it in your game, you just do x += velocity every frame right?

Acceleration is just the change in velocity over a given time.

So acceleration is delta A / T.

And to use it in your game, you just do velocity += acceleration every frame!

So instead of setting velocity, you can do:

if(key[KEY_UP]) acceleration += 0.1;
if(key[KEY_DOWN]) {acceleration -= 0.1; if(acceleration < 0)acceleration =0;}
velocity += acceleration;

//You can also make the car naturally slow down when you're not applying accel
velocity -= 0.01; 
if(velocity < 0)velocity = 0;

Lastly, FORCE is an acceleration. From newtons equation, force = mass * acceleration.

So if you factor in that a car has a mass, you can simulate a heavy or lighter car, and a beefier, or smaller engine easily.

Your engine is making a force. (Technically a force that changes per speed, which is why you have gears in a car, but you don't need that just yet!)

Your car has a mass. (mass is weight divided by earth's gravity)

So instead of directly applying acceleration, you apply FORCE. So if F=ma, then we can rearrange that to be a=f/m, acceleration is force divided by mass. Which passes the sanity test of "car accelerates faster if we reduce the mass, or increase the engine force."

float engine_force = 10; //or whatever
float car_mass = 1;

//...

if(key[KEY_UP])
{
acceleration += engine_force / car_mass; //a = f/m
}

That's it!

Now what about naturally slowing down? There's rolling resistance, and wind resistance. Rolling resistance is a function of the friction of everything rolling. Rolling resistance scales linearly with speed. Wind resistance is the same thing, but is the square of speed.

//rolling resistance
float rolling_resistance_factor = .1; //Some number. Best to just make one up.
float rolling_resistance_force = velocity * rolling_resistance_factor;

car_acceleration -= rolling_resistance_force / car_mass; //that's it!

//wind
float coefficient_of_drag = .01; //can look these up on wikipedia!
float drag_force = (velocity * velocity) * coefficient_of_drag;

car_acceleration -= drag_force / car_mass; //that's it!

I think it just looks more complex than it is.

Velocity is just your speed (or change in position over time)
Acceleration is just the change in velocity over time.

The rest of the stuff is just additional "factors" to determine the changes to apply to acceleration and velocity over time when you want to speed up or slow down. It makes it feel more "natural".

In a racing game, you may actually make the car mass, or engine mass variable based on each car. More powerful cars may have larger engines, with greater mass, making their enertia greater, causing their acceleration to change slower.

You could even apply "resistance" to steering (to object.angle, so you'd have an object.angle_accelleration or resistance something like that). Compare old american muscle cars or dragsters that have a very hard time turning, to rice rockets or super cars that can turn on a dime.

Velocity is a vector. A vector has a magnitude, and an angle. The magnitude determines how much is added each frame (of velocity, or position), and the angle determines what direction.

[edit]

You want the velocity, and acceleration updates to occur every logic event, typically after you've processed input. So with no mass or anything like that, you get something like this:

 if(ev.type == ALLEGRO_EVENT_TIMER)
{
if(keys[UP]) acceleration += .1;
if(keys[DOWN]) acceleration -= .1;
if(keys[LEFT]) whiteCar.angle += .1;
if(keys[RIGHT]) whiteCar.angle -= .1;

whiteCar.velocity += acceleration;
whiteCar.x += cos(whiteCar.angle)*whiteCar.velocity;
whiteCar.y += sin(whiteCar.angle)*whiteCar.velocity;
}

You shouldn't need a backward and forward functions because negative velocity goes backwards!

Whoops, I did write that reversed. Thanks for noticing it.

Velocity is velocity over delta-T, or V/dt. Delta means change. So the amount that time changes between two points.

For some reason, I can't edit that post anymore.

Maximum velocity should still work fine, just make sure it's checked after the acceleration is applied:

velocity += acceleration;
if(velocity > max_velocity)velocity = max_velocity;
if(velocity < -(max_reverse_velocity))velocity = -max_reverse_velocity;

Yes. They all mean the same thing in this context, but can occasionally mean other things in other contexts.

You need to apply drag by multiplying your velocity by a set factor. Use something close to 1.00 or so and only adjust the velocity by the drag when a timer is fired and after applying acceleration and not when decelerating (as that has its own coefficient). Also, acceleration is usually constant but you can increase it with time if a key is held down, like pushing harder on a gas or brake pedal.

Example :
If you update 60 times per second, then your drag factor will be applied exponentially, so you have to take care to calculate how much you will slow down in a second. Say you are using a drag factor of 0.99. (0.99)^60 is about 0.55. That means your car would lose 45% of its velocity every second. Anything less than .95 and you will probably stop almost immediately.

Thanks for the info guys, very informative.

At first, I just wanted something that 'did the job' but now that I've started I want to delve deeper into the physics.

Unfortunately for me, maths has never been my strong point so I'm battling to take this all in.

Bruce, you were right. I hadn't included the velocity heading in my IF KEY UP statement!

Now, my car will roll to a stop when moving in a forward direction, but stops dead in a reverse direction. It will be a logical error but I have tired eyes and cannot identify it.

My MoveCar() function is:

Also, I would like to apply something a little more realistic than object.velocity -=0.2 to roll my car to a stop. Rolling resistance or wind resistance? OR something else?

OK very close now!

The car rolls to a stop in both directions now, yey.

But there's a little flaw that's annoying me.

If I press a direction key, say UP, release before I reach maxSpeed and then quick press the opposite direction key, the car gains a sudden boost in acceleration in the original direction.

Sorry, I think I confused you I wanted you to think about what your existing code would do in those four states, to help you see that your second 'if' would always be entered if reached, and your third 'if' would never even be reached. It wasn't my intention to get you to write separate 'if' blocks for all those four states.

Chris has largely done the work for you, but you will have to read through it and add some missing blocks where commented. When I talked about lines you might be repeating unnecessarily, I meant the ones with sin and cos in them, which you can see Chris only has once.

Hope we've helped