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

[logan3]

I was wondering how speed, velocity, acceleration and anything with a $\Delta t$ in the denominator are defined at $\Delta t=0$? Other than approximating with limits, aren't they undefined?

Ex: ${\vec{v_{avg}} = \frac{\vec{s}}{\Delta t}}$, at t = 0 $\Rightarrow {\vec{v_{avg}} = \frac{\vec{s}}{0}} \Rightarrow {\vec{v_{avg}}} = und.$

${\vec{a}} = \frac{\vec{v_{f}}-\vec{v_{i}}}{\Delta t}$, at t = 0 $\Rightarrow {\vec{a}} = \frac{\vec{v_{f}}-\vec{v_{i}}}{0} \Rightarrow {\vec{a}} = und.$

Thank-you

 

[mfb]

None of those quantities are defined with a finite Δt. They are defined as derivatives, which avoids that issue by taking a mathematical limit.

 

[ChrisVer]

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 $\Delta t \rightarrow 0$ you have:

$a(t_0)= lim_{\Delta t \rightarrow 0} \frac{ u(t_{0}+ \Delta t) - u(t_{0})}{\Delta t}$
You can expand the $u(t_{0}+ \Delta t)$ around t0:
$u(t_{0}+ \Delta t) \approx u(t_{0})+ \Delta t \frac{du(t)}{dt}|_{t=t_{0}}$
(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 $\Delta t \rightarrow 0$ and so you see only the first power would survive-the rest would have $\frac{(\Delta t)^{2}}{\Delta t} \rightarrow 0$ in that limit).

thus:

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

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.