In the position in the figure, the spring is at its natural length. The block of mass m is given a velocity \(v_o\) towards the vertical support at t = 0. The coefficient of friction between the block and the surface is given by $\mu = \alpha x$, where $\alpha$ is a positive constant and x is the position of the block from its starting position. The block comes to rest for the first time at x, which is : 

\(v_o \sqrt{\frac{m}{k+\alpha .m g}}\)

Work-Energy Theorem : 

\(\frac{1}{2}mv^2_o = mg \alpha \int_o^x dx + \frac{1}{2}kx^2\)

Solving, we get : \(x = v_o \sqrt{\frac{m}{k+\alpha .m g}}\)