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Errors in Transverse and Longitudinal Accelerations

  1. Oct 17, 2015 #1
    Let's say you know all 3 cartesian components of a particle's velocity and all 3 for it's acceleration.

    You can split the acceleration vector into two vectors, one parallel (longitudinal) to the velocity vector and one perpendicular (transverse) to the velocity vector.

    Then, I found the x-components of both acceleration vectors in terms of the 6 variables listed at the start. That would be the component of each acceleration vector parallel to the x-axis. I used the dot product to derive it.

    As you can see below, I multiplied the transverse-x acceleration component by gamma and the longitudinal-x acceleration component by gamma cube. These are the formulas for the longitudinal and transverse masses.

    I can get a formula for the x-component of the Force.


    In order to get the second formula, I differentiated the x-component of the 3-momentum with respect to time.

    I expected both equations to be equivalent, but try as I might, I can't make them equal.

    Can you help me spot the error(s) in my formulas?
  2. jcsd
  3. Oct 17, 2015 #2


    Staff: Mentor

  4. Oct 19, 2015 #3
    Sorry, it didn't help me at all.

    Anyway, I've discovered my careless mistake. It is supposed to be multiplied, not divided, by gamma cube for the second equation.
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