Starting from the equation:
[tex]E=\frac{1}{2}mv^2[/tex]
Multiply both members by 2 to cancel out the factor of 1/2 on the right member of the equation:
[tex]\begin{gathered} \Rightarrow2E=2(\frac{1}{2}mv^2) \\ \Rightarrow2E=mv^2 \end{gathered}[/tex]
Divide both members by m to cancel out the factor of m on the right member of the equation:
[tex]\begin{gathered} \Rightarrow\frac{2E}{m}=\frac{mv^2}{m} \\ \Rightarrow\frac{2E}{m}=v^2 \end{gathered}[/tex]
Take the square root of both member of the equation to get rid of the exponent of v:
[tex]\begin{gathered} \Rightarrow\sqrt[]{\frac{2E}{m}}=\sqrt[]{v^2} \\ \Rightarrow\sqrt[]{\frac{2E}{m}}=v \end{gathered}[/tex]
Therefore, the equation rearranged so that it is solved for v is:
[tex]v=\sqrt[]{\frac{2E}{m}}[/tex]
