**QUESTION**

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Read each question carefully. Show all your work for each part of the question. The parts within the question may not have equal weight. Students will hang rings from the ceiling to demonstrate the trajectory of a projectile. A launcher will send a small ball into the air at a velocity and angle that will allow the ball to pass through all the rings, as shown in the figure. The horizontal position $x$ and vertical position $y$ of the ball at different points along the trajectory must be determined so that the rings can be placed at the correct positions. The students set the launcher for a launch angle of $\theta=40^{\circ}$ and measure the range of the projectile in order to determine its launch speed. The following data are collected. \begin{tabular}{|l|l|} \hline & Range \\ \hline Trial 1 & $8.92 \mathrm{~m}$ \\ \hline Trial 2 & $8.96 \mathrm{~m}$ \\ \hline Trial 3 & $9.10 \mathrm{~m}$ \\ \hline Trial 4 & $9.07 \mathrm{~m}$ \\ \hline Trial 5 & $9.16 \mathrm{~m}$ \\ \hline \end{tabular} (a) Derive an equation for the vertical position $y$ of the projectile during its flight. Express your answer in terms of $x$, $v_{0}, \theta$, and physical constants, as appropriate.