$a=25$

please do not copy previous answers

i$)$ Calculate numerically the maximum and minimum values that the height $y(t),$ velocity $v(t),$ and acceleration $a(t)$ of a ball with a mass of $m=0\mathrm{.}5kg$ thrown vertically upward direction. $(20$ points$)$ The ball moves under air resistance proportional to the square of its speed, with a resistance coefficient given as $c_d={25}_{\frac{?}{?}}1000(k\frac{g}{m}).$ It starts from position $y(0)=0$ with an initial velocity of $v(0)=5\frac{{25}_m}{s}.$ The gravitational acceleration is taken as $g=9\mathrm{.}81\frac{m}{s^2}.$ The equations of motion for the ball are given below:

$\frac{dv}{dt}=-g-\frac{c_d}{m}v|v|(1)$

$\frac{dy}{dt}=v(2)$

Equation $(1)$ shows the velocity$-$dependent change in acceleration and can be solved using the first$-$order forward finite difference method. The velocity values obtained here can then be used in numerical integration $($trapezoidal method$)$ to find the corresponding y values, as seen in Equation $(2).$

ii$)$ How long $($in seconds$)$ does it take for the ball to fall back to the ground? $(10$ points$)$

iii$)$ Calculate the terminal velocity the ball can reach when it rises to the highest point and falls back. At what second and at what height $($in meters$)$ does it reach half of this velocity? $(20$ points$)$ You may use electronic spreadsheets for your calculations. You can take the finite difference time interval as $t=0\mathrm{.}01s.$