# Thread: [SOLVED] Prove max distance angle of shooting a cannon

1. ## [SOLVED] Prove max distance angle of shooting a cannon

In class we proved that the best angle (ignoring air resistance) for shooting a cannon at a height of 0 is $45 deg$.

How would i prove what the best angle (less than or greater than 45) is if the cannon is raised to some Height, $H$, where $H>0$. I dont need the specific angle i just need to prove it is less than or greater than 45 degrees. I think its less than but I am not sure.

Here are the initial values of everything.
V=velcoity in m/s that the ball is shot at
$\theta$=angle (which we are trying to optimize)
H=starting height (greater than 0)
X=distance in terms of t
Y=Height in terms of t

$X= Vcos(\theta)t$
$Y= -4.9t^2 + Vsin(\theta)t + H$

From here i would normally find where $Y=0$.
So i used the quadratic equationa and got:

$\frac
{-fsin(\theta)t + \sqrt{f^2 sin^2(\theta) t^2 - 9.8t^2 h}}{-9.8t^2}
$

or the minus solution.

From here i am not sure what the next step would be to solve for t or even if this last step helps.

2. $= 0$

$
-fsin(\theta)t + \sqrt{f^2sin^2(\theta)t^2 - 9.8t^2h} = 0
$

$
f^2 sin^2(\theta)t^2 + f^2sin^2(\theta)t^2-9.8t^2h = 0
$

$
2f^2sin^2(\theta)t^2 - 9.8t^2h = 0
$

factor out $2t^2$:
$
2t^2(f^2sin^2(\theta)-4.9h)=0
$

t=0 or....here is where i end

3. I looked so hard for a solution...Thanks!