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Constantly accelerating rocket algebra problem

  1. Mar 21, 2016 #1

    Fek

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    1. The problem statement, all variables and given/known data
    • Rocket is accelerating constantly. Let S' be instantaneous rest frame of rocket and S be frame in which rocket is observed moving at velocity v.

    2. Relevant equations
    Given: $$ dv = dv' (1 - v^2) $$

    Must prove:
    $$ \frac{dv}{dt} = \frac{dv'}{dt'} (1 - v^2)^{\frac{3}{2}} $$

    3. The attempt at a solution

    So differentiate given equation with respect to t and use chain rule to get in terms of t'

    $$ \frac{dv}{dt} = \frac{dt'}{dt} * \frac{d}{dt'}[dv'(1 - v^2)] $$
    We also know
    $$ \frac{dt'}{dt} = (1 - v^2)^{\frac{1}{2}} $$
    as t' is proper time.

    We also have:
    $$ \frac{d}{dt'} (dv' (1 - v^2) = \frac{dv'}{dt'} (1 - v^2) $$

    We have the answer! Except this final step isn't right because v is a function of t' as well and chain rule must be used?
     
  2. jcsd
  3. Mar 21, 2016 #2

    PeroK

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    Science Advisor
    Homework Helper
    Gold Member

    If you have a functional equation, you can differentiate it. If you have an equation involving infinitesimal differentials, you can't differentiate it. Instead, you can divide by another infinitesimal differential.
     
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