\begin{problem*}{8.48}
A bullet of mass $m$ is fired horizontally into a block of mass $M$
initially at rest at the edge of a frictionless table of height $h$
(Fig. P8.48).  The bullet remains in the block, and after impact the
block lands a distance $d$ from the bottom of the table.  Determine
the intial speed of the bullet.
\end{problem*} % problem 87.48

\begin{solution}
Breaking the problem up into two parts (like problem 18), call the
point before the collision $A$, the point just after the collision $B$
and the point when the block-bullet hits the floor $C$.

From $B$ to $C$ is a standard projectile motion problem, which we'll
solve for the horizontal velocity $v_B$ of the block-bullet at point
$B$.  Because $v_B$ is purely horizontal (the \ihat direction), we'll
use the vertical (\jhat) direction to find the time it took the ball
to fall.
\begin{align}
 y_f = -h &= \frac{1}{2}a t^2 + v_{y0}t + y_0 = \frac{-g}{2}t^2 \\
 t &= \sqrt{\frac{2h}{g}} \\
 x_f = d &= v_B t + x_0 = v_B t \\
 v_B &= \frac{d}{t} = d\sqrt{\frac{g}{2h}} \\
\end{align}

Now conserving momentum back to $A$
\begin{align}
 P_B = (m+M) v_B &= P_A = m v \\
 v &= \frac{m+M}{m} v_B = \ans{\frac{m+M}{m}d\sqrt{\frac{g}{2h}}}
\end{align}
\end{solution}
