- #1

black_hole

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## Homework Statement

We are given L = 1/2mv

^{2}- 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 = -1/2gt

^{2}+ 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 = 1/2gt

^{2}- vozt = 0 when t = (z * v0z)/g = τ ?

(I'm not explicitly given v0z...)

c) S = ∫Ldt = 1/2m∫v

^{2}dt - mg∫zdt

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