$(15$ points$)$ By accident, a large plate is dropped and breaks into three pieces. The pieces fly apart parallel to the floor. As the plate falls, its momentum has only a vertical component, and no component parallel to the floor. After the collision, the component of the total momentum parallel to the floor must remain zero, since the net external force acting on the plate has no component parallel to the floor. Using the data shown in the drawing, find the mass of piece $1,m_1.$

A$.0\mathrm{.}681$ kg

B$.0\mathrm{.}373$ kg

C$.0\mathrm{.}592$ kg

D$.0\mathrm{.}557$ kg