Help with a mechanical lagrangian problem

[black_hole]

Homework Statement

We are given L = 1/2mv² - mgz.

a) Find the equations of motion.
b) Take x(0) [vector] = 0; v(0) [vector] = v0 [vector] ; v0z > 0 and find x(τ) [vector] and v(τ) [vector], such that z(τ) = 0;  τ≠0.
c) Find S.

Homework Equations

Euler-Lagrange equation and definition of action.

The Attempt at a Solution

a) Using the Euler-Lagrange equation:

∂L/∂x - d/dt(∂L/∂v) = 0
-mg - d/dt(mv) = 0
-mg = d/dt(mv)
F = -dp/dt

b) If F = -dp/dt, then a = -g. So;

v = -vt + v0 and x = -1/2gt² + v0t

Hmmm, that's where I'm stuck; I'm not sure how to implement the last two constraints. Maybe if z(τ) = 0 but τ≠0 then z = 1/2gt² - vozt = 0 when t = (z * v0z)/g = τ ?
(I'm not explicitly given v0z...)

c) S = ∫Ldt = 1/2m∫v²dt - mg∫zdt

If my above expressions for v and z are correct then I presume I can use those tot find the integrals...?

 

[I like Serena]

Hi black_hole!

You seem to be mixing up x and z.

This is a ball being thrown up (v0_z > 0).
They want to know when the ball is back on the ground...
Or rather where the ball is at that time and what its velocity is as a function of its initial velocity v0.