Maximizing the height of a bullet in cylindrical coordinates

[Qbit42]

Homework Statement

A gun can fire shells in any direction with the same speed v0. Ignoring air resistance and using cylindrical polar coordinates with the gun at the origin and z measure vertically up, show that the gun can hit any object inside the surface

z = $$\frac{v_{0}^{2}}{2g}$$ - $$\frac{g\rho^{2}}{2v_{0}^{2}}$$

Homework Equations

$$\phi$$ is fixed so any derivative terms can be neglected, making Newtons equations:

F$$_{\rho}$$ = m$$\frac{d^{2}\rho}{dt^{2}}$$

F$$_{z}$$ = m$$\frac{d^{2}z}{dt^{2}}$$

The Attempt at a Solution

I know how I should tackle this problem, but I can't get started. I want to use Newton's laws to solve for z(t) and $$\rho$$(t). Differentiate z(t) to solve for max height at t$$_{final}$$. Then solve $$\rho$$(t$$_{final}$$) for t$$_{final}$$($$\rho$$) and use that to find z($$\rho$$).

I have no idea how to solve Newtons equations in this case, it seems like F$$_{z}$$ = F$$_{g}$$ and I have no idea what expression to use for F $$_{\rho}$$

Edit: I do not know why my subscripts are being interpreted as superscripts but I can't get it to stop.

 

[diazona]

Without getting into too much detail, the subscript/superscript thing is because of a feature of the browser layout. Don't mix LaTeX code with regular letters; it's better to put the entire equation or expression in LaTeX.

$$F_\rho = m\frac{\mathrm{d}^2\rho}{\mathrm{d}t^2}$$

and $z(t)$ etc.

Anyway: you've correctly identified that the force in the z (vertical) direction is the force of gravity. What is the net force acting in the ρ (horizontal) direction? This part of the problem is simple projectile motion, except that the coordinates are labeled ρ and z instead of x and y as you might be used to.

Once you've found the functions $z(t)$ and $\rho(t)$, you will need to combine them to find $z(\rho)$, and then differentiate - but not to find the max height. You should instead be looking at the direct 3D distance between the projectile and the launch point.

 

[Qbit42]

Thanks for the clarification on the superscript problem. I honestly have no idea what force would be acting on the bullet in the horizontal direction, its been a long time since I've done any classical mechanics and I always hated it (still do).

As to the differentiation part, If I have z($$\rho$$) then I guess I'd want to differentiate with respect to $$\rho$$ or $$\phi$$ to maximize 3D distance.

Edit: Do I have to split the initial velocity into components based on angle between $$\rho$$ and z? Like $$V_{oz}$$ = $$V_{0}$$Sin($$\theta$$) and $$V_{o\rho}$$ = $$V_{0}$$Cos($$\theta$$). Although I don't see how that would help since I need an expression for the time derivative of $$V_{0}$$ and those are just constants.