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

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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.

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