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

question 104

Essay

Provide an appropriate response.
-Derive the equations
$$\begin{array} { l } x = x _ { 0 } + \frac { v _ { 0 } } { k } \left( 1 - e ^ { - k t } \right) \cos \alpha \\y = y _ { 0 } + \frac { v _ { 0 } } { k } \left( 1 - e ^ { - k t } \right) \sin \alpha + \frac { g } { k ^ { 2 } } \left( 1 - k t - e ^ { - k t } \right)\end{array}$$
by solving the following initial value problem for a vector $\mathbf { r }$ in the plane.
$$\begin{aligned}\text { Differential equation } \frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm { dt } ^ { 2 } } & = - g \mathbf { j } - \mathrm { k } \mathbf { v } = - \mathrm { g } \mathbf { j } - \mathrm { k } \frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } \\\text { Initial conditions: } \quad \mathrm { r } ( 0 ) & = \mathrm { x } _ { 0 } \mathbf { i } + \mathrm { y } _ { 0 } \mathbf { j } \\& \frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } ( 0 ) = \mathbf { v } _ { 0 } = \left( \mathrm { v } _ { 0 } \cos \alpha \right) \mathbf { i } + \left( \mathrm { v } _ { 0 } \sin \alpha \right) \mathbf { j }\end{aligned}$$
The drag coefficient $\mathrm { k }$ is a positive constant representing resistance due to air density, vo and $\alpha$ are the projectile's initial speed and launch angle, and $g$ is the acceleration of gravity.

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