Question

Some high school physics students dropped a ball and measured the distance fallen (in centimeters) at various times (in seconds) after its release. If you have studied physics, then you probably know that the theoretical relationship between the variables is distance $=490{\left(time\right)}^2$. A scatterplot of the students’ data showed a clear curved pattern. Which of the following single transformations should linearize the relationship? I. ${time}^2$II. $dis\tan c{e}^2$III. $\sqrt{dis\tan ce}$

(a) I only

(c) III only

(e) I and III only

(b) II only

(d) I and II only

Short answer

The single transformation that should linearize the relationship is option (e) I and III only.

Step-by-step solution

Given information

The theoretical relationship between the variables is the distance$=490{\left(time\right)}^2$.

Explanation

Equation theoretical model

Distance $=490{\left(time\right)}^2$

Equation linear model

$dis\tan ce=a+b\times time$

If we employ the transformation, the equation of the linear model and the equation of the theoretical model become identical ${\left(time\right)}^2$.

Take the square root of each side of the theoretical model's equation:

$\sqrt{dis\tan ce}=\sqrt{490}\times time$

The equation of the linear model and the theoretical model are then equivalent if we use the transformation$\sqrt{dis\tan ce}$

Thus I and III only are transformations that should linearize the relationship.