I've tried using the accelrometer a couple of years ago just for this purpose, and I can tell you that the noise will
far exceed any change caused by movement. But, if you are set on it, you should research some better methods for double integration; the traditional method has way too much error on its own.
https://en.wikipedia.org/wiki/Simpson%27s_rule
https://en.wikipedia.org/wiki/Gaussian_quadrature
The idea is to use a polynomial to approximate the function, rather than a flat rectangle or trapazoid as one learns in high school calculus. If you have any questions, I can help. I don't see the accelerometer working without something like this, since the error for the integration alone will be too high, let alone error from the reading (it is only meant to tell you the direction of gravity, nothing more!)
Also, the mentor on my team who deals with such things has had trouble getting the kit ultrasonic sensors to go more than ~20 feet, iirc. You will need to find some high quality ones for that to work.