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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Definitions:

Isotonic

A solution that has the same solute concentration as another solution, preventing the movement of water across a semipermeable membrane.

Osmosis

The spontaneous movement of solvent molecules through a semi-permeable membrane from a region of lower solute concentration to one of higher solute concentration.

Translocation

The movement of substances within an organism or the transfer of genes or segments of chromosomes from one location to another.

Transpiration

The process by which moisture is carried through plants from roots to small pores on the underside of leaves, where it changes to vapor and is released into the atmosphere.