Canon shooting area in polar coordinates

[prehisto]

Homework Statement

hi,guys.
The directions of shooting e=cos$\alpha$cos$\varphi$i+cos$\alpha$sin$\varphi$j+sin$\alpha$k
0<$\varphi$<=2π;$\varphi$ -horizontally
$\alpha$[0,π];$\alpha$ is vertically
initial speed=v₀

I need to calculate the surface equation of canon shots (where it hits).
In other words equations of the surface which is made from canon hits.

Homework Equations

The Attempt at a Solution

My view is that i need to gain horizontal and vertical distance (r),and then i can get the desired equation.

So horizontal distance is r=t|v_xy|=tv₀|e_x+e_y|=tv₀(cos$\alpha$cos$\varphi$i+cos$\alpha$sin$\varphi$j)^1/2=tv₀cos($\alpha$)

So now i need height or in other words the distance in vertical direction(z -direction) or all together?.. I have hard time recalling use of equations of motion.
some help,please?

 

[BvU]

Is there a question in the problem statement ?
You'll find equations of motion here, or otherwise $\vec F = m\vec a$ is a good one too.

List what you need under 2 and use less shorthand than in

So horizontal distance is r=t|v_xy|=tv₀|e_x+e_y|=tv₀(cos$\alpha$cos$\varphi$i+cos$\alpha$sin$\varphi$j)^1/2=tv₀cos($\alpha$)

 

[prehisto]

Is there a question in the problem statement ?
You'll find equations of motion here, or otherwise $\vec F = m\vec a$ is a good one too.

List what you need under 2 and use less shorthand than in

Im sorry,i edited the original post,tried to be more specific.

 

[BvU]

Let me try to translate: you want to calculate where a cannon ball lands that is shot from a cannon that is placed on the x-y plane in an empty flatland with no atmosphere, and gravity acceleration g in the -z direction, as a function of $\phi, \theta$ v₀ and g.
That right ? Something like http://www.rabidgeek.net/physics-applets/projectile-motion/ ?

Did you find something useful among the equations of motion? Or here or here ?

 

[prehisto]

Let me try to translate: you want to calculate where a cannon ball lands that is shot from a cannon that is placed on the x-y plane

Yes,and the direction of cannon is described by $\vec{e}$=cos$\alpha$cos$\varphi$i+cos$\alpha$sin$\varphi$j+sin$\alpha$k

Because 0<$\varphi$<2pi and $\alpha$[0,pi].Result is not one spot where it would land,but a surface which I am looking for.

Gravity g=-g$\vec{k}$

Did you find something useful among the equations of motion? Or here or here ?

Yes,I found something useful,I think I could use d_v=v_vt-gt²/2

 

[BvU]

Yes, so d_v=0 has two solutions, one trivial at shooting off and one at "landing".
Bingo.