$1990$B$1.$ A bullet of mass $m$ is moving horizontally with speed $v_0$ when it hits a block of mass $100m$ that is at rest on a horizontal frictionless table, as shown above. The surface of the table is a height $h$ above the floor. After the impact, the bullet and the block slide off the table and hit the floor a distance $x$ from the edge of the table. Derive expressions for the following quantities in terms of $m,h,v_o,$ and appropriate constants:

a$.$ the speed of the block as it leaves the table

b$.$ the change in kinetic energy of the bullet$-$block system during impact

c$.$ the distance $x$

Suppose that the bullet passes through the block instead of remaining in it$.$

d$.$ State whether the time required for the block to reach the floor from the edge of the table would now be greater, less, or the same. Justify your answer.

e$.$ State whether the distance $x$ for the block would now be greater, less, or the same. Justify your answer.