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This is actually a physics experiment. I've collected a set of data

Distance and time it takes for a cart to move down an incline plane. From here, I've calculated the average time (AVERAGE(all t's for that D)) and then velocity ($\frac{D}{t_{avg}}$).

I am then asked to find an "even closer approximate" to the instantaneous velocity of the cart as it passes the midpoint of the incline plane. Also providing the uncertainty (STDERR). How do I do that, isn't D/Average Time the best I can do?

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    You are computing average velocity. Instantaneous velocity is different. It is the velocity at a particular instant in time. In this case it is at the midpoint down the plane.2012-09-09

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You should find a good fit with a linear regression of $t$ vs. $\sqrt D$. At least that would match a law $s(t)=a t^2$. If $D$ is reached after time $t_0$ (that is $s(t_0)=D$) then $\frac 12 D$ is reached after time $t_1=\frac1{\sqrt2}t_0$ and at that monent the instantaneous velocity is $v(t_1)=s'(t_1) = 2 a t_1=\sqrt 2 a t_0$. With $D_0=at_0^2$ we find $v(t_1)=\sqrt 2 \frac{D_0}{t_0}$

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    Hmm, I don't really get you at "*then $\frac{1}{2}D$ is reached after time $t_1 = \frac{1}{\sqrt{2}} t0$ ...*".2012-09-09