QUESTION

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# 2. A spring with a spring force given by the formula: $\mathrm{F}=-\alpha x^{\frac{1}{2}}$, where $\mathrm{x}$ is the distance from equilibrium that the spring is compressed or stretched and $\alpha$ is a constant. A group of physics students used the spring on an experiment to investigate the elastic properties of the spring. They recorded the collected data in the table below. \begin{tabular}{|l|l|} \hline Force (N) & $\begin{array}{l}\text { Compression } \\ (\mathrm{m})\end{array}$ \\ \hline 2.5 & 0.01 \\ \hline 3.4 & 0.02 \\ \hline 4.2 & 0.03 \\ \hline 5.2 & 0.04 \\ \hline 5.5 & 0.05 \\ \hline 5.9 & 0.06 \\ \hline 6.5 & 0.07 \\ \hline 7.2 & 0.08 \\ \hline 7.5 & 0.09 \\ \hline \end{tabular} a. On the diagram below, make a graph that can be used to determine the constant A. Be sure to label the $x$ and $y$ axes with the appropriate variable. b. Find the constant $\alpha$ from the graph. c. Develop an expression for the potential energy that can be stored in the spring. d. The spring is used in a spring gun placed horizontally at the edge of a $0.75 \mathrm{~m}$ tall table. The spring is compressed to a potential energy of $1.8 \mathrm{~J}$. It then launches a 0.5 $\mathrm{kg}$ dart. Find the maximum distance from the table where a dart projected from the gun hits the floor.  