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Speed, Velocity, Acceleration, etc. at t=0

  1. Sep 13, 2014 #1
    I was wondering how speed, velocity, acceleration and anything with a [itex]\Delta t[/itex] in the denominator are defined at [itex]\Delta t=0[/itex]? Other than approximating with limits, aren't they undefined?

    Ex: [itex]{\vec{v_{avg}} = \frac{\vec{s}}{\Delta t}}[/itex], at t = 0 [itex]\Rightarrow {\vec{v_{avg}} = \frac{\vec{s}}{0}} \Rightarrow {\vec{v_{avg}}} = und.[/itex]

    [itex]{\vec{a}} = \frac{\vec{v_{f}}-\vec{v_{i}}}{\Delta t}[/itex], at t = 0 [itex]\Rightarrow {\vec{a}} = \frac{\vec{v_{f}}-\vec{v_{i}}}{0} \Rightarrow {\vec{a}} = und.[/itex]

    Thank-you
     
  2. jcsd
  3. Sep 13, 2014 #2

    mfb

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    2016 Award

    Staff: Mentor

    None of those quantities are defined with a finite Δt. They are defined as derivatives, which avoids that issue by taking a mathematical limit.
     
  4. Sep 13, 2014 #3

    ChrisVer

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    Gold Member

    Well as mfb pointed out, they are defined by derivatives, and thus they are instantaneous and not average.
    There is no average velocity/acceleration defined at a given time t=0, but in some interval Δt. So your formulas are wrong...

    For [itex]\Delta t \rightarrow 0[/itex] you have:

    [itex] a(t_0)= lim_{\Delta t \rightarrow 0} \frac{ u(t_{0}+ \Delta t) - u(t_{0})}{\Delta t}[/itex]
    You can expand the [itex] u(t_{0}+ \Delta t)[/itex] around t0:
    [itex] u(t_{0}+ \Delta t) \approx u(t_{0})+ \Delta t \frac{du(t)}{dt}|_{t=t_{0}}[/itex]
    (you could try to write more terms in the series so avoid the approximation symbol, but later on you are going to take the limit of [itex]\Delta t \rightarrow 0[/itex] and so you see only the first power would survive-the rest would have [itex] \frac{(\Delta t)^{2}}{\Delta t} \rightarrow 0 [/itex] in that limit).

    thus:

    [itex] a(t_0)= lim_{\Delta t \rightarrow 0} \frac{ u(t_{0})+ \Delta t \frac{du(t)}{dt}|_{t=t_{0}} - u(t_{0})}{\Delta t}=\frac{du(t)}{dt}|_{t=t_{0}} [/itex]

    As for a the acceleration (rate of change of velocity), a similar way is used for u, but with x instead of u (since it's the rate of change of position)

    But there is no distinction between derivatives and limits... derivatives are mathematically defined by the first limit I wrote...now if you prefer listening to the name "derivative" instead of "limit" it's up to you. But all derivative properties are proved by their definition as limits.
     
    Last edited: Sep 14, 2014
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