Closest model for relationship between velocity and distance Which is the model that best shows the relationship between velocity and distance: 


*

*$v \propto d$

*$v \propto d^2$

*$v^2 \propto d$

*$v^2 \propto d^2$



What I think is from: 
$$v=at \Rightarrow t=\frac{t}{a}$$
Then substitute into: 
$$\begin{aligned}
d &= \frac{1}{2} at^2 \\
&= \frac{1}{2} \frac{v^2}{a^2} \\
&= \frac{1}{2} \frac{v^2}{a} \\
\end{aligned}$$
So can I say closest model should be $d \propto v^2$? But my experimental data shows that $d^2 \propto v$. That is by plotting a least linear squares fit with excel and using the $r^2$ value. Could use LINEST() too. 

Is my data wrong then? 
 A: If all you have is the data there's not a lot more you can do. I doubt the difference in fit between the $v:d$ and $v:d^2$ fits is significant.
I note that neither fit goes through the origin, which makes me suspect that neither fit captures the physics behind the data.
You need to have a look at your system and see if you can write down some (maybe approximate) equation to model it's behaviour. I would guess that you have some system with friction/drag involved, so the acceleration starts out high and falls with increasing speed. In that case if you can make a guess at the relationship between velocity and drag you can write down an approximate equation of motion and then fit that to the data.
A: I would first advice to use a second order polynomial as a fitting model instead of a linear one. And indeed to use r to determine which model is the best.
You can not use your formula for any processes but for a free fall (or constant acceleration). For other cases there is no reason that v=a*t and d=v^2/2*a. For diffudion process for instance the velocity is proportional to sqrt(t)
