A solid wooden door, $90$ cm wide by $2\mathrm{.}0$ m tall, has a mass of $35$ kg $.$ It is open and at rest. A small $500-$g ball is thrown perpendicular to the door with a speed of $20\frac{m}{s}$ and hits the door $60$ cm from the hinged side, causing it to begin turning. The ball rebounds along the same line with a speed of $16\mathrm{.}0\frac{m}{s}$ relative to the ground. How much energy is lost during this collision?

A$)4\mathrm{.}8$ J

B$)15$ J

C$)16$ J

D$)30$ J

E$)13$ J