Add equation number in array

amsmath has several environments designed to number each row, or as here number the group as a whole. Do not use array for such display constructs the spacing and style are not designed for that.

\documentclass[12pt,a4paper]{scrartcl}
\usepackage{amsmath}
\usepackage{amssymb,amsthm}
\begin{document}

\textbf{Conservation of energy}
\begin{equation}
\begin{split}
  E_{\mathrm{kin}}                  &= E'_{\mathrm{kin}}                   \\
  m_1 v_1^2 + m_2 v_2^2             &= m_1 {v'}_{1}^{2} + m_2 {v'}_{2}^{2} \\
  m_1 (v_1^2 - {v'}_1^2)            &= m_2 ({v'}_2^2 - v_2^2)              \\
  m_1 (v_1 + {v'}_1) (v_1 - {v'}_1) &= m_2 ({v'}_2 + v_2) ({v'}_2 - v_2) 
\end{split}
\end{equation}

\textbf{Conservation of momentum}  
\begin{equation}
\begin{split}
  p                  &= p'                      \\
  m_1 v_1 + m_2 v_2  &= m_1 {v'}_1 + m_2 {v'}_2 \\
  m_1 (v_1 - {v'}_1) &= m_2 ({v'}_2 - v_2)
\end{split}
\end{equation}


\end{document}

You could do it this way:

\documentclass[12pt,a4paper]{scrartcl}
\usepackage{amsmath}
\usepackage{amssymb,amsthm}
\begin{document}
\begin{align}
  & \textbf{Conservation of energy} 
   & E_{kin}                          &= E'_{kin}                         \notag\\
  && m_1 v_1^2 + m_2 v_2^2            &= m_1 {v'}_{1}^{2} + m_2 {v'}_{2}^{2}    \\
  && m_1 (v_1^2 - {v'}_1^2)           &= m_2 ({v'}_2^2 - v_2^2)           \notag\\
  && m_1 (v_1 + {v'}_1) (v_1 - {v'}_1)&= m_2 ({v'}_2 + v_2) ({v'}_2 - v_2)\notag\\[2ex]
  & \textbf{Conservation of momentum}
   & p                                &= p'                               \notag\\
  && m_1 v_1 + m_2 v_2                &= m_1 {v'}_1 + m_2 {v'}_2                \\
  && m_1 (v_1 - {v'}_1)               &= m_2 ({v'}_2 - v_2)               \notag
\end{align}
\end{document}

You can use alignedat, but some manual work is needed for sharing the column widths across the two parts; we identify the widest columns in the global alignment and set three dimensions before doing the display.

\documentclass[12pt,a4paper]{scrartcl}
\usepackage{amsmath,mathtools,amssymb,amsthm}

\usepackage{lipsum} % just for the example

\newlength{\izA}\newlength{\izB}\newlength{\izC}
\newcommand{\mmb}[3]{\mathmakebox[#1][#2]{\displaystyle#3}}

\begin{document}

\lipsum*[2]
%%% settings for the display
\settowidth{\izA}{\textbf{Conservation of momentum}}%
\settowidth{\izB}{$\displaystyle m_1 (v_1 + {v'}_1) (v_1 - {v'}_1)$}%
\settowidth{\izC}{$\displaystyle m_2 ({v'}_2 + v_2) ({v'}_2 - v_2)$}%
%%%
\begin{gather}
\begin{alignedat}{2}
&\mmb{\izA}{l}{\textbf{Conservation of energy}}
   & \mmb{\izB}{r}{E_{\mathrm{kin}}}   &= \mmb{\izC}{l}{E'_{\mathrm{kin}}}    \\
  && m_1 v_1^2 + m_2 v_2^2             &= m_1 {v'}_{1}^{2} + m_2 {v'}_{2}^{2} \\
  && m_1 (v_1^2 - {v'}_1^2)            &= m_2 ({v'}_2^2 - v_2^2)              \\
  && m_1 (v_1 + {v'}_1) (v_1 - {v'}_1) &= m_2 ({v'}_2 + v_2) ({v'}_2 - v_2)
\end{alignedat}
\\
\begin{alignedat}{2}
&\textbf{Conservation of momentum}
   & \mmb{\izB}{r}{p}                  &= \mmb{\izC}{l}{p'}                   \\
  && m_1 v_1 + m_2 v_2                 &= m_1 {v'}_1 + m_2 {v'}_2             \\
  && m_1 (v_1 - {v'}_1)                &= m_2 ({v'}_2 - v_2)
\end{alignedat}
\end{gather}
\lipsum[3]

\end{document}

Alternative solution with split, so we need less measuring.

\documentclass[12pt,a4paper]{scrartcl}
\usepackage{amsmath,mathtools,amssymb,amsthm}

\usepackage{lipsum} % just for the example

\newlength{\lefthandside}
\newcommand{\leftbox}[2][]{%
  \makebox[\lefthandside][s]{#1\hfill$\displaystyle#2$}%
}

\begin{document}

\lipsum*[2]
%%% settings for the display
\settowidth{\lefthandside}{%
  \textbf{Conservation of momentum}%
  \quad
  $\displaystyle m_1 (v_1 + {v'}_1) (v_1 - {v'}_1)$%
}%
%%%
\begin{align}
\begin{split}
\leftbox[\textbf{Conservation of energy}]
  {E_{\mathrm{kin}}}                        &= E'_{\mathrm{kin}} \\
\leftbox{m_1 v_1^2 + m_2 v_2^2}             &= m_1 {v'}_{1}^{2} + m_2 {v'}_{2}^{2} \\
\leftbox{m_1 (v_1^2 - {v'}_1^2)}            &= m_2 ({v'}_2^2 - v_2^2)              \\
\leftbox{m_1 (v_1 + {v'}_1) (v_1 - {v'}_1)} &= m_2 ({v'}_2 + v_2) ({v'}_2 - v_2)
\end{split}
\\
\begin{split}
\leftbox[\textbf{Conservation of momentum}]
  {p}                                       &= p'                   \\
\leftbox{m_1 v_1 + m_2 v_2}                 &= m_1 {v'}_1 + m_2 {v'}_2             \\
\leftbox{m_1 (v_1 - {v'}_1)}                &= m_2 ({v'}_2 - v_2)
\end{split}
\end{align}
\lipsum[3]

\end{document}