Derivation of kinetic energy

  1. Feb 23, 2012 #1
    Hi everyone,

    There are 2 things I do not understand in the derivation of kinetic energy from work:

    (1) W = [itex]\int[/itex][itex]\vec{F}[/itex](t).d[itex]\vec{r}[/itex](t)=

    (2) m.[itex]\int[/itex][itex]\frac{d\vec{v}(t)}{dt}[/itex].d[itex]\vec{r}[/itex](t)=

    (3) m.[itex]\int[/itex]d[itex]\vec{v}[/itex](t).[itex]\frac{d\vec{r}(t)}{dt}[/itex]=

    (4) [itex]\frac{m}{2}[/itex].(v(t1) - v(t0))[itex]^{2}[/itex]

    Question I: I don't understand why you can just change the division like that between (2) and (3). I know multiplication and division have the same order but they are calculated from left to right. So why can you do that from (2) to (3)?

    Question II: I don't understand why the change in kinetic energy between t1 and t0 is sometimes written as [itex]\frac{1}{2}[/itex].m.v(t1)[itex]^{2}[/itex] - [itex]\frac{1}{2}[/itex].m.v(t0)[itex]^{2}[/itex]
    E.g. like in

    After (4), it should be [itex]\frac{m}{2}[/itex].v(t1)[itex]^{2}[/itex] - m.v(t1).v(t0) + [itex]\frac{m}{2}[/itex].v(t0)[itex]^{2}[/itex] , right?
    Last edited: Feb 23, 2012
  2. jcsd
  3. Feb 23, 2012 #2

    Doc Al

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    Staff: Mentor

    No. Your step (4) is incorrect.
  4. Feb 23, 2012 #3
    I'm sorry, I don"t understand what I am doing wrong going from (3) to (4). Can you elaborate please?
  5. Feb 23, 2012 #4

    Doc Al

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    Staff: Mentor

    ∫v dv = v2/2 → v12/2 - v02/2
  6. Feb 24, 2012 #5

    Philip Wood

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    Gold Member

    Re your QI...
    I respect your unwillingness to regard [itex]\frac{d\vec{v}}{dt}[/itex] as one thing divided by another, but you might be happier to do this with [itex]\frac{Δ\vec{v}}{Δt}[/itex], and then associate Δt with Δ[itex]\vec{r}[/itex] to make [itex]\vec{v}[/itex]. Only then take it to the limit.

    I don't suppose this is rigorously valid, but I suspect this is how a lot of physicists justify this sort of transposition.
  7. Feb 24, 2012 #6
    Thnx for your responses.
  8. Feb 25, 2012 #7
    Question I: Because a derivative is equivalent to a division of differentials. And then by usual algebraic rules you can move the denominator.

    Question II: It is not sometimes, but always, because that is the result of the integration of (3). You own result (4) is incorrect.
    Last edited: Feb 25, 2012
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