Motion equation with gravity

When I do this I sum all the gravity vectors each frame and then apply that to the target object.

To solve the timing issue you refer to I simply scale the composite gravity effect against the length of the interval between frames.

I am using this method for a large number of dynamic objects and it's pretty efficient. If you wanted to get really clever you could use some AltiVec or something on well organized sets of values.

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Original post by doctorsixstring Given the motion equation (d = v*t + 0.5*a*t2), I can find a vehicle's position after a given passage of time with a given starting velocity and constant acceleration.

As you've stated, keep in mind this formula only works when acceleration remains constant. Obviously a game with constant acceleration would be pretty boring, so this formula isn't exactly correct - it's still correct for the duration of one time-step, because in our simulations we update the input forces (i.e. update the acceleration) only once per frame. So for the duration of a frame we can say that the acceleration remains constant.

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Now I want to throw in some gravity wells (planets, black holes, etc.). How can I calculate the final position of a missile or ship at a given time, with an initial velocity, acceleration, and one or more sources of gravity?

The above forumula remains the same, you just also want to add in the formula f=m*a (or acceleration = force / mass). Gravity is just another force being applied to your objects, so it is going to affect the acceleration which is fed into your first formula.

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I know I could simply apply gravity to the ship's position each game tick, but that would yield different results depending on the length of each tick. I would really like a solution that yields identical results regardless of the change in time.

Welcome to the world of numerical integration ;)
Games (almost?) always use approximate methods for integration, such as Euler's method or verlet integration. Usually these approximations are stabilised by enforcing a fixed time-step.

For example, always use a delta-time value of 10ms. If 35ms have passed since the last update, then you run the integrator 3 times, and add 5 onto a 'leftover' variable which will be added onto next frames delta value.

Alternatively, my Verlet link above describes a "time corrected" method that remains stable even when the time-step varies.

Algebraically, no I don't think you can do that. It seems like it would be tantamount to solving the n-body problem.

Thanks for the replies, guys.

I'll go ahead and start working on using fixed-length time steps, and running one or more updates per frame, depending on frame length. I'll apply gravity, ship thrust, and other external forces as a summed acceleration vector at the start of each frame.

Since we've established that it would be extremely difficult to create a single formula, I won't be able to use it to solve for vehicle thrust with a desired final position.

So in order to calculate power and angle of launched missiles, it sounds like I'll need to use brute force to calculate a solution. I will calculate each frame's update logic ahead of time and see where the missile will end up. If it misses, I'll keep re-running the trajectory calculation for a slightly changed power/angle until the missile hits. This seems pretty computationally expensive, but I don't see any other way.