A \(6.0-\mathrm{kg}\) bowling ball is held \(10 \mathrm{~mm}\) above a mattress and then released from rest and allowed to fall, sinking into the mattress. Model the mattress as a single spring with spring constant \(k=500 \mathrm{~N} / \mathrm{m}\).

(a) Calculate the ball's kinetic energy, the gravitational potential energy of the system (ball, spring, and Earth), and the elastic potential energy at the following compressions: \(0,0.050 \mathrm{~m}, 0. 10 \mathrm{~m}\), \(0.15 \mathrm{~m}, 0. 20 \mathrm{~m}\).

(b) What is the maximum compression of the mattress? Ignore the inertia of the spring.