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 $c_d=0\mathrm{.}028(k\frac{g}{m}).$ It starts from position $y(0)=0$ with an initial

velocity of $v(0)=140\frac{m}{s}.$ The gravitational acceleration is taken as $g=9\mathrm{.}81\frac{m}{s^2}.$ The equations

of motion for the ball are given below:

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

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

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 $t=0\mathrm{.}01s.$

i$)$ A vertical force $P=28kN$ acts on the end of a cantilever beam with a rectangular cross$-$section

width of $28cm$ and a height that changes linearly. Divide the beam length into $20$ equal parts

and calculate the deflection and rotation at the midpoint of the beam using the finite difference

method. $L=4m,h_0=30cm,h_L=15cm,E=10$GPa