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.