As the equation* environment doesn't permit line breaks, consider using a multline* environment. And, instead of creating a multi-line surd expression with a \sqrt instruction, I suggest you use (...)^{1/2} notation.
Two additional comments about your code: Do try to use \left and \right less frequently, and don't needlessly encase various terms in curly braces.

\documentclass[11pt,a4paper,twocolumn]{article}
\usepackage{amsmath} % for 'multline*' env.
\begin{document}
...
\hrule % just to illustrate column width
\begin{multline*}
| \dot{r} | = \bigl\{
- e^{-t} ( \cos t + \sin t )^{2} \\
+ e^{-t} ( \cos t + \sin t )^{2}
+ (-e^{-t})^{2} \bigr\}^{1/2}
\end{multline*}
\end{document}
alignedenvironment and, the symbol&and\\. Have a look here – gnikit Apr 10 '17 at 17:45alignat*frommathtools(oramsmath) package? – Apr 10 '17 at 17:56breqn? – Alexander Wittmann Jun 12 '22 at 14:45