Closest model for relationship between velocity and distance

Question

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

• $v \propto d$
• $v \propto d^2$
• $v^2 \propto d$
• $v^2 \propto d^2$

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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?

Answer 1

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.

Answer 2

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)