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V dot (dv/dt) = (0.5)*(d/dt)*(v^2) ?

  1. Sep 13, 2010 #1
    This is not a homework question, but a derivation in my class which I can't get around.

    1. The problem statement, all variables and given/known data
    Step1 [tex]F=ma[/tex]
    Step2 [tex]\vec{F} = m\frac{d\vec{v}}{dt}[/tex]
    Step3: Multiply both side by v [tex]\vec{F}.\vec{v} = m\vec{v}.\frac{d\vec{v}}{dt}[/tex]
    Step4 [tex]Power = \frac{d}{dt}\frac{1}{2}m\vec{v}.\vec{v}[/tex]
    Step5 [tex]Power = \frac{d}{dt}\frac{1}{2}mv^{2} = \frac{dK}{dt}

    2. Relevant equations

    3. The attempt at a solution
    I just can't figure out where did the [tex]\frac{1}{2}[/tex] come from in step 4.

    Please help & thanks in advance!
  2. jcsd
  3. Sep 13, 2010 #2


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    Homework Helper


    So that d/dt(v2)=2v*dv/dt

    They just wrote v2 as v.v
  4. Sep 13, 2010 #3
    Aha, I see.
    Or more elaborately:

    [tex]\frac{d}{dt}v^{2} = \frac{dv}{dt}\frac{d}{dv}v^{2} = \frac{dv}{dt}2v[/tex]

    Thanks rock!
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