Arduino+memsic to measure velocity? Please help.

I have been thinking about building a project to measure the speed of a elevator in the up and down directions. I currently use a rotary tachometer to measure the the speed In FPM and adjust a hydraulic valve to speed up or slow down the elevator to contract speed. Ex (100-125-150FPM)

So my project idea if possible is to use an arduino-accelerometer combo to measure the vertical velocity by running a program to calculate the- initial velocity+(acceleration x time)= final velocity.
In this case the initial velocity would be zero because I would start my measurements from a stand still, then I would calibrate the accelerometer data to approximate a base line measure. From This point I would have to create a timed event of acceleration to calculate final velocity.

My question to you is this possible?

My mock calculations started with

Initial velocity+(acceleration x time) = final velocity
0 + .357071 m/s2 x 4 sec =.51m/s or 100 feet per minute

Depends how accurate you want the result. If you want a velocity that is +/- 5% I don't think an accelerometer will give you that. It might be good for a larger error range.