Solve varepsilon_{c}=1/2mv^2

$$\varepsilon _ { c } = \frac { 1 } { 2 } m v ^ { 2 }$$

Solve for m

$$\left\{\begin{matrix}m=\frac{2\epsilon _{c}}{v^{2}}\text{, }&v\neq 0\\m\in \mathrm{R}\text{, }&\epsilon _{c}=0\text{ and }v=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}mv^{2}=\epsilon _{c}$$

The equation is in standard form.

$$\frac{v^{2}}{2}m=\epsilon _{c}$$

Divide both sides by \frac{1}{2}v^{2}.

$$\frac{2\times \frac{v^{2}}{2}m}{v^{2}}=\frac{2\epsilon _{c}}{v^{2}}$$

Dividing by \frac{1}{2}v^{2} undoes the multiplication by \frac{1}{2}v^{2}.

$$m=\frac{2\epsilon _{c}}{v^{2}}$$

Solve for v

$$\left\{\begin{matrix}v=\sqrt{\frac{2\epsilon _{c}}{m}}\text{; }v=-\sqrt{\frac{2\epsilon _{c}}{m}}\text{, }&\left(\epsilon _{c}\geq 0\text{ and }m>0\right)\text{ or }\left(\epsilon _{c}\leq 0\text{ and }m<0\right)\\v\in \mathrm{R}\text{, }&\epsilon _{c}=0\text{ and }m=0\end{matrix}\right.$$