Bob B of mass m at rest is hanging vertically from the ceiling via...

Since the collision is elastic and the masses of $A$ and $B$ are equal, they will exchange their velocities upon collision.

Velocity of bob $B$ just after the collision, $v = 10$ ms$^{-1}$.

By conservation of mechanical energy for bob $B$ during its upward swing:

$\dfrac{1}{2} m v^2 = m g h$

Substituting the given values ($v = 10$ ms$^{-1}$, $g = 10$ ms$^{-2}$):

$\dfrac{1}{2} \times m \times (10)^2 = m \times 10 \times h$

$50 = 10 h$

$h = 5$ m.

Since $h < L$ (where $L = 10$ m), the string does not become slack and the bob reaches a maximum height of $5$ m.

Answer: $5$