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.