How can I factor momentum into my space sim?

Question

I am trying my hand at creating a simple 2d physics engine right now, and I'm running into some problems figuring out how to incorporate momentum into movement of a spaceship.

If I am moving in a given direction at a certain velocity, I am able to currently update the position of my ship easily (Position += Direction * Velocity). However, if the ship rotates at all, and I recalculate the direction (based on the new angle the ship is facing), and accelerate in that direction, how can I take momentum into account to alter the "line" that the ship travels? Currently the ship changes direction instantaneously and continues at its current velocity in that new direction when I press the thrust button. I want it to be a more gradual turning motion so as to give the impression that the ship itself has some mass.

If there is already a nice post on this topic I apologize, but nothing came up in my searches. Let me know if any more information is needed, but I'm hoping someone can easily tell me how I can throw mass * velocity into my game loop update.

Answer 1

It looks like you have Position and Direction as vectors and Velocity as a scalar?

If so just change Velocity into a vector as well and then do something like this:

Force = Direction * Power
Acceleration = Force / Mass
Velocity += Acceleration * ElapsedTime
Position += Velocity * ElapsedTime

Direction being a unit length vector giving the heading of your ship. Power being the amount of thrust you want to add (Zero when no acceleration is needed) ElapsedTime is the time between your game Update calls to keep movement smooth even when your framerate is changing

If you want to know more about it you can look up Euler Integration

Answer 2

Instead of manipulating the velocity directly, add another variable: acceleration, that gradually, smoothly alters the velocity:

From classical physics (Newton's laws of motion):

a =  F/m  // a Force will result in a smaller acceleration on objects with more mass
v += a    // acceleration is the rate of change in velocity
p += v    // velocity is the rate of change in position    

where:

a = acceleration, v = velocity, p = position
F = force, m = mass

(Note m is the only scalar value; F, a, v, and p are all 2D or 3D vectors)

Also, there are actually two directions:

1. The direction the ship is facing. Force from engines is applied along this vector.
2. The direction the ship is actually moving (due to momentum). This is the velocity vector v.

I answered a related question on StackOverflow: 2D Spaceship movement math. There is some sample code in that answer.