Data Linearization Non$-$Linear Spring A

A block of mass $m$ is initially in contact with an uncompressed non$-$linear spring of negligible mass. The block is pushed back along a frictionless surface a distance $x,$ as shown below, compressing the spring. The block is then released from rest. At the unstretched length of the spring, the block leaves the spring with velocity $v$ and runs through a photogate that records its velocity. The spring force of this spring is related to its deformation such that $F_S=-C{x}^2$ where $C$ is a positive unknown constant.

$-$cs

A$.$ Determine the functionality between $x$ and $v$ using energy method:

B$.$ The experiment is executed, varying the compression $x.$ The mass $m$ is $125$ grams and the data are shown below. Construct a best$-$fit line to the data.

$\backslash$table$[[$trial$,x(m),v(\frac{m}{s}),v^{\frac{2}{3}}(\frac{m}{s}{)}^{\frac{2}{3}}$