Energy Balance Spring Slider system

[SquareCircles]

I am currently looking facing a problem concerning a spring slider system.
The system consists of a electro engine that pulls on a spring with a constant velocity. The spring is attached to a block that sticks to the surface, but after a certain force is applied (by the spring) starts to slip and move a certain distance. I want to calculate the average velocity during the slip event. To get an as close to realistic situation as possible the force applied by the electric engine during the slip event also has to be taken into accound.
So far I came up with this:
F*v_{pull velocity electric engine}=0.5kx_{displacement spring}^2+0.5m(v_block^2-mgx_{displacement block}(mu)
Now I have been told that I can solve the problem by integration over time. But I am rather confused as to how I can do this.

 

[Simon Bridge]

You cannot pull with a constant velocity - you can move at a constant velocity.
To move at a constant velocity while pulling one a spring means that the force supplied by the engine varies with it's position.

Fv is the rate that energy is arriving via the engine - it is energy per unit time.
Everything on the other side is energy. So your units don't match.
x in the equation 0.5kx^2 is not the displacement of the spring, it is the extension of the spring.

You can probably work it out by forces if you track the position of the engine as $x_e(t)=x_i+vt$, the position of the block is $x_b(t)$, and the unstretched length of the spring is $s$ ... and put $x_b(0)=0$. Draw a free-body diagram for the block, and apply Newton's laws. Don't forget there will be static as well as kinetic friction.

The motion can get quite complicated - the mass will oscillate unless critically damped, and may catch and stick sometimes if the static friction is high enough.