Consider a mass m attached to a spring with natural length l, hanging vertically under the action of gravity mgk (where the unit vector k is pointing downwards) and a constant friction force F : F0k. (a) Find the equilibrium point xe of the mass, write the equation of motion, and show that the motion of the particle is governed by the fundamental equation of simple harmonic motion. (b) Assume the particle is released from the spring when it has height h above ground and initial velocity v0. Let y be the height above ground of the particle (note that the orientation of the axis is now opposite of z used in point (a)). Write the equation of motion (under the action of gravity and the friction force F). Solve them for the given initial condition and show that v2 = v5+2 m (0) Upon entering the ground (y : 0) with velocity v1, the particle is subject to a constant friction force F1 where F1 > 0 is a constant. Calculate the distance d travelled by the particle into the ground in terms of F1 and m,g, h, F0 introduced in point (b)