Surprisingly often I'm faced with estimating the position of something that moves according to Newtonian laws of motion in one dimension and with random acceleration.
A fast and pretty accurate way of doing this is to use a Kalman filter. I've finally gotten around to implementing a specialized version of this general method, dedicated to my simple estimations needs. You are welcome to use it (it's HPL licensed).
Have a look at the image if you want to see how it performs in a sample case.