Example $2$

Consider the trajectory of a ball thrown from the top of a $30$ meter building at time $t=0$ sec$,$ with initial velocity vector vec$(v)=(5,5),$ meaning $5\frac{m}{sec}$ horizontal speed and $5\frac{m}{sec}$ vertical speed. Let $(x(t),y(t))$ be the parametric curve tracking its horizontal and vertical position at $tsec,$ assuming the initial position $(x(0),y(0))=(0,30).$

B

$($a$)$ Give the formula for the horizontal position $x(t),$ moving with constant velocity.

$($b$)$ Give the formula for the vertical position $y(t)$ as a ballistic equation

$y(t)=y_0+v_{0}t+\frac{1}{2}g{t}^2$

where $y_0$ is the initial position, $v_0$ is the initial velocity, and $g=-10\frac{m}{se{c}^2}$ is the acceleration of gravity $($minus means downward$).$

$($c$)$ When does the ball hit the ground? Compute its terminal velocity vector vec$(v)(t)=(x^{\text{'}}(t),y^{\text{'}}(t))$ and terminal speed $\sqrt[\phantom{2}]{(x^{\text{'}}(t){)}^2+(y^{\text{'}}(t){)}^2}.$ Sketch the trajectory and the velocity arrows at the initial and terminal points.