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What happens to Newton's 2nd law if there is a changing mass?

  1. Aug 14, 2010 #1
    I understand that force is equal to the time derivative of momentum, or d(mv)/dt. Then what happens if the velocity is constant and only the mass is changing. Does this mean there will be a force. If so, in what direction since I am assuming it is still a vector. Also, what if the mass and velocity are changing, does this make some kind of "2nd order" force?
    Any help would be appreciated.
  2. jcsd
  3. Aug 14, 2010 #2


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    Gold Member

    So what you have now is [tex] \frac{d}{dt}(m(t)v(t))[/tex]. This is simply a product rule, nothing beyond that. You'll simply have a more complicated problem.

    For example, if you had a simple harmonic oscillator, your 2nd law would become:

    [tex]\frac{d}{{dt}}(m(t)\dot x(t)) = - kx(t)[/tex]

    which upon doing the differentiation simply gives

    [tex]\dot m(t)\dot x(t) + m(t)\ddot x(t) = - kx(t)

    Of course, you'll need information on the functional relationship of m with respect to t to solve this. I haven't given this much thought but that does not look like an easy problem
  4. Aug 14, 2010 #3
    I don't understand how you can have a change in mass. Newton didn't anticipate mass annihilation, so I'm not sure what becomes of the law.
  5. Aug 14, 2010 #4
    You can have a system in which you just ignore a part of the mass.For example you ignore the fuel that was ejected in a rocket if you are interested only in the motion the rocket. F=dp/dt so both mass and velocity can vary.This does not mean that mass is annihilated only that you can sometimes ignore some of it.
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