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Help with a mechanical lagrangian problem

  1. Jan 10, 2013 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    Euler-Lagrange equation and definition of action.

    3. 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...?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 10, 2013 #2

    I like Serena

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    Homework Helper

    Hi black_hole! :smile:

    You seem to be mixing up x and z.

    This is a ball being thrown up (v0z > 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.
     
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