Provide an Appropriate Response By Solving the Following Initial Value Problem for a Vector

Essay

Provide an appropriate response.
-Derive the equations
$\begin{array} { l } x = x _ { 0 } + \left( v _ { 0 } \cos \alpha \right) t \\y = y _ { 0 } + \left( v _ { 0 } \sin \alpha \right) t - \frac { 1 } { 2 } g ^ { 2 }\end{array}$
by solving the following initial value problem for a vector $\mathbf { r }$ in the plane.
Differential equation $\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm { dt } ^ { 2 } } = - \mathrm { gj }$ Initial conditions:
$$\begin{array} { l } \mathbf { r } ( 0 ) = \mathrm { x } _ { 0 } \mathbf { i } + \mathrm { y } _ { 0 } \mathbf { j } \\\left. \frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } ( 0 ) = \mathrm { v } _ { 0 } \cos \alpha \right) \mathbf { i } + \left( \mathrm { v } _ { 0 } \sin \alpha \right) \mathbf { j }\end{array}$$