A $500$ g mass on a horizontal spring oscillates with a period of $2\mathrm{.}12$ s and an amplitude of $75\mathrm{.}0$ cm$.$ During one oscillation, at the instant the spring is at maximum extension, the mass explodes into two pieces. One piece is shot forward, away from the spring, at $2\mathrm{.}00$ m$/$s$.$ The second piece, which remains attached to the spring, recoils at $3\mathrm{.}00$ m$/$s$.$

A$)$ After the explosion, what is the period of the oscillation?

B$)$ After the explosion, what is the amplitude of the oscillation?