Problem source

In the $Z$--universe, a star of mass $M$
suddenly blows up, and the fragments, with various initial speeds, 
start to move away from the centre of mass $G$ which may be
regarded as a fixed point. In the  subsequent motion the 
acceleration of each fragment is directed towards $G$.
Moreover, in accordance with the laws of physics of the $Z$--universe, 
there  are positive constants 
$k_1$, $k_2$ and $R$ such that when a fragment is at a distance $x$
from $G$, the magnitude of its acceleration  is $k_1x^3$ if
$x < R$ and  is $k_2x^{-4}$ if $x \ge R$.
The initial speed of a fragment is denoted by $u$.
\begin{questionparts}
\item
For $x < R$, write down a differential equation for the speed $v$,
and hence determine $v$ in terms of $u$, $k_1$ and $x$ for $ x < R$.  
\item
 Show that if $u < a$, where $2a^2=k_1 R^4$, 
then the fragment does not reach a distance $R$ from $G$.
\item Show that if $u \ge b$, 
where 
$
6b^2= 3k_1R^4  + 4k_2 /R^3,
$ 
then from the moment
of the explosion the fragment is always moving away from $G$.
\item If $a < u < b$, determine in terms of 
$k_2$, $b$ and $u$ 
the maximum distance from $G$ attained by the fragment.
\end{questionparts}