6. A small object $P$, of mass $m _ { 0 }$, is projected vertically upwards from the ground with speed $U$. As $P$ moves upwards it picks up droplets of moisture from the atmosphere. The droplets are at rest immediately before they are picked up. In a model of the motion, $P$ is modelled as a particle, air resistance is assumed to be negligible and the acceleration due to gravity is assumed to have the constant value of $g$. When $P$ is at a height $x$ above the ground, the combined mass of $P$ and the moisture is $m _ { 0 } ( 1 + k x )$, where $k$ is a constant, and the speed of $P$ is $v$.
\begin{enumerate}[label=(\alph*)]
\item Show that, while $P$ is moving upwards

$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( v ^ { 2 } \right) + \frac { 2 k v ^ { 2 } } { ( 1 + k x ) } = - 2 g$$

The general solution of this differential equation is given by $v ^ { 2 } = \frac { A } { ( 1 + k x ) ^ { 2 } } - \frac { 2 g } { 3 k } ( 1 + k x )$,\\
where $A$ is an arbitrary constant. Given that $U = \sqrt { 2 g h }$ and $k = \frac { 7 } { 3 h }$
\item find, in terms of $h$, the height of $P$ above the ground when $P$ first comes to rest.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M5 2017 Q6 [12]}}