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Help on derivation of work

  1. Feb 21, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm working through a derivation for work in Thornton's Classical Dynamics but I'm stuck at one step.
    [tex] \begin{align}
    \vec{F} \cdot d\vec{r} &= m\frac{d\vec{v}}{dt} \cdot \frac{d\vec{r}}{dt}dt = m\frac{d\vec{v}}{dt} \cdot \vec{v}dt \\ &= \frac{m}{2}\frac{d}{dt}(\vec{v}\cdot \vec{v})dt \end{align}[/tex]

    I'm having trouble getting from the last equality on the first line to the second line.

    2. Relevant equations

    3. The attempt at a solution
    I don't know what the mathematical reasoning is.
  2. jcsd
  3. Feb 21, 2013 #2
    I don't get it too. Might be a mistake.
  4. Feb 21, 2013 #3
    Last edited: Feb 21, 2013
  5. Feb 21, 2013 #4


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    Science Advisor

    You can't say that because you haven't defined "v^2". Since v is a vector, what you really mean is [itex]\vec{v}\cdot\vec{v}= ||\vec{v}||^2[/itex].
    But, the basic idea is correct:
    [tex]\frac{d(\vec{v}\cdot\vec{v})}{dt}= 2\vec{v}\cdot\frac{d\vec{v}}{dt}[/tex]

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