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Homework Help: Particle Disintegration: Equation trouble

  1. Jul 14, 2010 #1
    Hi, I'm reading about particle disintegration at the moment and there's a step I don't follow.

    I've got the following equation:

    [tex]\tan\theta=\frac{v_0\sin\theta_0}{v_0\cos\theta_0+V} [/tex] where [tex] \theta [/tex] is the resultant angle in the Laboratory system and [tex] \theta_0 [/tex] is the resultant angle in the Centre of Mass system.

    Also given is [tex]v=V+v_0[/tex] which are respectively the velocity of a resulting particle in the L system, the velocity of the primary particle in the L system, and the velocity of the resulting particle in the C system.

    Solving for [tex]\cos\theta_0[/tex] one should obtain

    [tex]\cos\theta_0=-\frac{V}{v_0}\sin^2\theta \pm \cos\theta\sqrt{1-\frac{V^2\sin^2\theta}{v_0^2}}[/tex]

    but I've gotten [tex]\cos\theta_0=\frac{V}{v_0}(\cos\theta-1)+\cos\theta[/tex]

    from the substitution [tex]\sin\theta_0=\sin\theta\left(\frac{V+v_0}{v_0}\right)[/tex] which seems geometrically sound.

    Can anyone see where that other equation comes from for [tex]\cos\theta_0[/tex]?

    Thanks.
     
  2. jcsd
  3. Jul 14, 2010 #2

    vela

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    Are you sure the equation you started with applies to this situation? It's valid for non-relativistic cases, but particle decay typically involves relativistic speeds.
     
  4. Jul 15, 2010 #3
    Yes, in the book, it follows on immediately, with no mention of relativistic effects.
    I think it's just some algebraic reworking of the equation that I'm not seeing.
     
  5. Jul 15, 2010 #4

    vela

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    From the equation

    [tex]\tan\theta=\frac{v_0\sin\theta_0}{v_0\cos\theta_0+V}[/tex]

    I'm inferring that the angles are measured relative to the direction of V, the velocity of the primary particle in the lab frame. Your second equation from equating the y-components in the two frames is wrong. It should be

    [tex]v_0\sin\theta_0=\sqrt{(v_0\cos\theta_0+V)^2+(v_0\sin\theta_0)^2}\sin\theta[/tex]

    The equation relating the velocities should be

    [tex]\vec{v}=\vec{v}_0+\vec{V}[/tex]

    which is a vector equation, so you can't just add the magnitudes of v0 and V to get the magnitude of v.

    To derive the other equation, start with the tan θ equation, square it, and rewrite sin2 θ0 in terms of cos θ0. You'll get a quadratic equation in cos θ0.
     
  6. Jul 15, 2010 #5
    Excellent! I got it!

    That was some marathon of a reworking - three pages and an hour later...

    Thanks vela for your help. :)
     
  7. Jul 15, 2010 #6
    Excellent! I got it!

    That was some marathon of a reworking - three pages and an hour later...

    Thanks vela for your help.
     
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