Particle moving on a conical surface

[Drajcoshi]

Homework Statement

A particle moves under the action of gravity on a conical surface z^2 = 4(x^2+ y^2),
z ≥ 0, where z is the vertical axis. For initial position r = (1, 0, 2) and initial velocity
ṙ = (0, 2, 0) find the extremal values of z along the trajectory. Take g = 10.

Homework Equations

I really have not a clue how to type the equation on this site but have uploaded the work out on pdf. will appreciate if anyone can shed some light on this. thanks

 

[TSny]

Hello, Drajcoshi. Welcome to Physics Forums!

I'm not following your set up of polar coordinates. Did you really want to set $z = \rho$?

I think you can solve this problem with just application of conservation laws. Besides energy, can you think of anything else that's conserved?

 

[Drajcoshi]

At this point I am we'll confused, how would you do it? Really appreciate your help. I think I completely messed up the calculation.

 

[TSny]

There's another quantity that's conserved (hint: it's the z-component of some vector quantity).

With this quantity and energy you will be able to set up equations to determine max or min of z.

 

[Drajcoshi]

alright, did more calcultion and stuck don't know how to find the max and min of z. I have included the calculation, could you check it for me? thanks

https://www.dropbox.com/s/0qq6cyopw4d0bvj/photo.JPG

 

[TSny]

OK, so you have that the z-component of angular momentum as well as the total energy is conserved.

Try writing expressions for $L_z$ and $E$. You are only concerned with points of max or min $z$ and the expression for $L_z$ will simplify at those points.

 

[Drajcoshi]

this this correct? what happens next? sorry all this maths notation is so confusing since i am not a maths student. also don't know how to write using the equation editor.

 

[TSny]

I'm not really following your notation. If you let $v$ represent the speed of the particle, how would you express $E$ in terms of $v$ and $z$?

Can you also express $L_z$ in terms of $v$ and $z$ at a point of maximum or minimum $z$? (First express it in terms of $v$ and $\rho$ and then express $\rho$ in terms of $z$.)