What is the Speed of a Particle Falling Between Two Cones?

[khemist]

Homework Statement

Consider two hollow fixed cones (such as ice cream cones without the ice cream). They have a base radius R, slant height L,and a surface mass density σ. The cones are vertical, with their apexs touching (say, at the origin). A particle initially at rest falls in from infinity, along a perpendicular bisector line. What is its speed when it reaches the tip of the cones?

Homework Equations

F = GMm/r^2
V= GMm/r

The Attempt at a Solution

So I am trying to write an equation for the cone, where if I pick any arbitrary height on the cone, I would get the circumference at that point. I would then integrate over the height of the cone. However, I am having trouble coming up with such an equations. I know that the forces in the z and y direction will cancel, so the particle will be "pulled" towards the vertex of both cones.

Would it be possible to consider the cones as a point at their center of mass?

 

[I like Serena]

Hi khemist!

I don't really understand what your cones have to do with it...

At infinity a particle of mass m and speed zero would have energy zero (arbitrary choice).
At distance 6000 km from Earth (which is the radius of the earth), you can calculate the energy E=-GMm/6000km.
This will equal the increase in kinetic energy (1/2)mv^2, from which you can calculate the speed v...

Btw, note that the potential is V=-GM/r.