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A small ball of mass m starts at a point A with speed $$v_o$$ and moves along a frictionless track AB as shown. The track BC has coefficient of friction $$\mu$$. The ball comes to stop at C after travelling a distance L which is:
$$E_A = mgh + \frac{1}{2}mv_o^2$$
$$E_C = 0$$ (ball comes to stop)
$$W_{friction} = -f \cdot L = -\mu mgL$$
$$mgh + \frac{1}{2}mv_o^2 = \mu mgL$$ (The initial energy is dissipated by the work done by friction)
$$L = \frac{mgh}{\mu mg} + \frac{\frac{1}{2}mv_o^2}{\mu mg}$$
$$L = \frac{h}{\mu} + \frac{v_o^2}{2\mu g}$$
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