Solve y=y_{0}+v_{o}t+1/2at^2

$$y = y _ { 0 } + v _ { o } t + \frac { 1 } { 2 } a t ^ { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{2\left(tv_{o}+y_{0}-y\right)}{t^{2}}\text{, }&t\neq 0\\a\in \mathrm{C}\text{, }&y=y_{0}\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$y_{0}+v_{o}t+\frac{1}{2}at^{2}=y$$

Subtract y_{0} from both sides.

$$v_{o}t+\frac{1}{2}at^{2}=y-y_{0}$$

Subtract v_{o}t from both sides.

$$\frac{1}{2}at^{2}=y-y_{0}-v_{o}t$$

The equation is in standard form.

$$\frac{t^{2}}{2}a=y-y_{0}-tv_{o}$$

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

$$\frac{2\times \frac{t^{2}}{2}a}{t^{2}}=\frac{2\left(y-y_{0}-tv_{o}\right)}{t^{2}}$$

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

$$a=\frac{2\left(y-y_{0}-tv_{o}\right)}{t^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{2\left(tv_{o}+y_{0}-y\right)}{t^{2}}\text{, }&t\neq 0\\a\in \mathrm{R}\text{, }&y=y_{0}\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$y_{0}+v_{o}t+\frac{1}{2}at^{2}=y$$

Subtract y_{0} from both sides.

$$v_{o}t+\frac{1}{2}at^{2}=y-y_{0}$$

Subtract v_{o}t from both sides.

$$\frac{1}{2}at^{2}=y-y_{0}-v_{o}t$$

The equation is in standard form.

$$\frac{t^{2}}{2}a=y-y_{0}-tv_{o}$$

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

$$\frac{2\times \frac{t^{2}}{2}a}{t^{2}}=\frac{2\left(y-y_{0}-tv_{o}\right)}{t^{2}}$$

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

$$a=\frac{2\left(y-y_{0}-tv_{o}\right)}{t^{2}}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{\sqrt{2ay-2ay_{0}+v_{o}^{2}}-v_{o}}{a}\text{; }t=-\frac{\sqrt{2ay-2ay_{0}+v_{o}^{2}}+v_{o}}{a}\text{, }&a\neq 0\\t=-\frac{y_{0}-y}{v_{o}}\text{, }&a=0\text{ and }v_{o}\neq 0\\t\in \mathrm{C}\text{, }&a=0\text{ and }v_{o}=0\text{ and }y=y_{0}\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=\frac{\sqrt{2ay-2ay_{0}+v_{o}^{2}}-v_{o}}{a}\text{; }t=-\frac{\sqrt{2ay-2ay_{0}+v_{o}^{2}}+v_{o}}{a}\text{, }&\left(y\geq 0\text{ and }y_{0}=\frac{v_{o}^{2}}{2a}\text{ and }a>0\right)\text{ or }\left(a\neq 0\text{ and }y_{0}=\frac{v_{o}^{2}}{2a}\text{ and }y=0\right)\text{ or }\left(y\geq -\frac{v_{o}^{2}-2ay_{0}}{2a}\text{ and }y_{0}\leq \frac{v_{o}^{2}}{2a}\text{ and }a>0\right)\text{ or }\left(y_{0}\geq \frac{v_{o}^{2}}{2a}\text{ and }y\leq -\frac{v_{o}^{2}-2ay_{0}}{2a}\text{ and }a<0\right)\\t=-\frac{y_{0}-y}{v_{o}}\text{, }&a=0\text{ and }v_{o}\neq 0\\t\in \mathrm{R}\text{, }&a=0\text{ and }v_{o}=0\text{ and }y=y_{0}\end{matrix}\right.$$