The Initial Value Problem $( L - x ) \frac { d ^ { 2 } x } { d t ^ { 2 } } - \left( \frac { d x } { d t } \right) ^ { 2 } = L g , x ( 0 ) = 0 , x ^ { \prime } ( 0 ) = 0$

The initial value problem $( L - x ) \frac { d ^ { 2 } x } { d t ^ { 2 } } - \left( \frac { d x } { d t } \right) ^ { 2 } = L g , x ( 0 ) = 0 , x ^ { \prime } ( 0 ) = 0$ is a model of a chain of length $L$ falling to the ground, where $x ( t )$ represents the length of chain on the ground at time $t$ . The solution for $v = \frac { d x } { d t }$ in terms of $x$ is

A) $v = \sqrt { L g \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) }$
B) $v = \left( \operatorname { Lg } \left( L ^ { 2 } + ( L - x ) ^ { 2 } \right) \right) / ( L - x )$
C) $v = \left( \operatorname { Lg } \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) \right) / ( L - x )$
D) $v = \sqrt { L g \left( L ^ { 2 } + ( L - x ) ^ { 2 } \right) } / ( L - x )$
E) $v = \sqrt { L g \left( L ^ { 2 } - ( L - x ) ^ { 2 } \right) } / ( L - x )$