how to calculate phone's movement in the vertical direction from rest?

If you integrate the acceleration twice you get position but the error is horrible. It is useless in practice. Here is an explanation why (Google Tech Talk) at 23:20. I highly recommend this video.

Now, you do not need anything accurate and that is a different story. The linear acceleration is available after sensor fusion, as described in the video. See Sensor.TYPE_LINEAR_ACCELERATION at SensorEvent. I would first try a high-pass filter to detect sudden increase in the linear acceleration along the vertical axis.

I have no idea whether it is good for your application.

You can actually establish (only) the vertical position without measuring acceleration over time. This is accomplished by measuring the angle between the direction to the center of the earth, and the direction to the magnetic north pole.

This only changes (significantly) when the altitude (height) of the phone changes. What you do is use the accelerometer and magnetometer to get two float[3] arrays, treat these as vectors, make them unit vectors, and then the angle between any two unit vectors is arccos(AxM).

Note that's dot product ie. math.acos(A[0]*B[0]+A[1]*B[1]+A[2]*B[2]) Any change in this angle corresponds to a change in height. Also note that this will have to be calibrated to real units and the ratio of change in angle to height will be different at various longitudes; But this is a method of getting an absolute value for height; though of course the angle also becomes skewed when undergoing acceleration, or when there are nearby magnets :)