Find at what initial velocity can the projectile reach $8534m?$ Find velocity and $x$ value when projectile reach $8543m$, without using Graph Method.

Question

How can I solve this problem not graphically? If it is not possible, what more information do I need?

At what initial velocity can the projectile reach $8534m?$ Find velocity and $x$ value when the projectile reach $8543m$.

$$x(T) = v\cos(θ)T$$ $$y(T)= h + v\sin(θ)T + \frac12 gT^2$$

Given (with proper units)
$h = 83m\\θ = 89°\\g = -9.8m/sec^2$

Answer 1

$x$($t$)$=vcos(\theta)t$ $\rightarrow$ ① and $y(t)=h+vsin(\theta)t+\frac{1}{2}gt^2$ $\rightarrow$②

you want to know the initial velocity of a projectile when it reach a height$=8534$ which will be in this case the max height the projectile have reached this means that the velocity in vertical direction will be equal to zero , $v_y$$=0$

$\frac{dy}{dt}$$=0$,$\frac{dy}{dt}=vsin(\theta)+gt$$=0$,$vsin(\theta)=-gt$ $\rightarrow$③

the term $(vsin(\theta))$ will be equal to ($-gt$) at the instant of reacting that height then we can use this conclusion and substitute in ②

$8534=83-gt^2+\frac{1}{2}gt^2$ we can solve this quadratic eqation and $t=41.5 sec$

use that $t=41.5$ and substitute in ③ you will get that $v=406.79$ $m/sec$

for the $x$ value substitute by $t=41.5$ and $v=406.79$ in ① and you will get that $x=294.607$ $meter$