Answer the following questions with reference to one-dimensional collisions between two bodies.
a. (i) State the two physical conditions that must simultaneously hold for a collision to be perfectly elastic.
b. (ii) Define the coefficient of restitution e for a head-on collision.
c. (iii) Prove that if e=1, the total kinetic energy of the two-body system is the same before and after the collision without explicitly assuming conservation of kinetic energy.
d. (iv) Explain why, in a perfectly elastic collision, the kinetic energies of the individual bodies may change even though the total kinetic energy of the system remains constant.