Suppose that a parachutist wtih linear drage $($m$=70$kg$,$ c$=12\mathrm{.}5$ kg$/$s$)$ jumps from an airplane flying at an altitude of $200$ m with a horizontal velocity of $180\frac{m}{s}$ relative to the ground.

PLEASE LOOK AT SECOND IMAGE FOR REQUIREMENTS FOR ANSWERS. Thanks

$($a$)$ Write a system of four differential equations for $x,y,v_x=d\frac{x}{d}t$ and $v_y=d\frac{y}{d}t.$

$($b$)$ If the initial horizontal position is defined as $x=0,$ use Euler's methods with $t=1s$ to compute the jumper's position over the first $10$s$.$

$($c$)$ Develop plots of $y$ versus $t$ and $y$ versus $x.$ Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open.

$($d$)$ At what angle would the parachutist be traveling in the last whole second before impact?

Requirements for answers

$(1)$ Write the differential equations for x$,$y$,$v$\_($x$),$ and v$\_($y$)$ ;

$(2)$ Write the numerical difference equations for x$,$y$,$v$\_($x$)$ and v$\_($y$)$ based on the Euler's method

$(3)$ Solve for x$,$y$,$v$\_($x$),$ and v$\_($y$)$ analytically and numerically with an initial step size of $1$ s and

plot the results vs$.$ time t in the graphs $($in separate graphs: x only, y only, v$\_($x$)$ only and v$\_($y$)$ only$).$

Solve for the angle of the traveling when the parachutist hits the ground.

$(4)$ Change the time step to a half of the initial time step size and recalculate numerically and

plot the results in the same graph in $(3)$ ;

$(5)$ Discuss the difference and changes in the results when using different time step size.

Calculate the error of the numerical results by comparing the numerical results with the

analytical results and analyze the effect of the step size on the error $.$