AP

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Unit $3$ FRQs

$2.$

In an experiment, students are to calculate the spring constant $k$ of a vertical spring in a small jumping toy that initially rests on a table. When the spring in the toy is compressed a distance x from its uncompressed length $L_0$ and the toy is released, the top of the toy rises to a maximum height $h$ above the point of maximum compression. The students repeat the experiment several times, measuring $h$ with objects of various masses taped to the top of the toy so that the combined mass of the toy and added objects is $m.$ The bottom of the toy and the spring each have negligible mass compared to the top of the toy and the objects taped to it$.$

$($a$)$ Derive an expression for the height $h$ in terms of $m,x,k,$ and fundamental constants.

With the spring compressed a distance $x=0\mathrm{.}020m$ in each trial, the students obtained the following data for different values of $m.$

$\backslash$table$[[m(kg),h(m)$