This question is a long freeresponse question. Show your work for each part of the question. V0 \\\\ +x Sphere' '\\ \\ +y >\"\\' Floor _\\ Motion of a Sphere Students are studying the two-dimensional motion of objects as they move through the air. Specically, they are examining the behavior of a sphere that is launched horizontally from a location above the oor with an initial velocity 1:0 in the +9: direction, as shown in the gure. The students assume that the positive directions are along the sphere's initial velocity for horizontal motion and downward for vertical motion. The horizontal displacement of the object from its starting point is 1:, and the vertical displacement of the object from its starting point is y. One of the students derives an equation for y in terms of a: and other quantities. After examining the equation, the student claims that y is proportional to :52. (a) Derive an equation for the vertical coordinate yof the sphere as a function of at. 1:0, and physical constants. as appropriate. (b) Design an experimental procedure that the students could use to test the claim that y is proportional to 1'2. in the table below, li. quantities and associated symbols that would be measured in your experiment and the equipment used to measure them. Also list the equipment that would be used to measure each quantity. You do not need to ll in every row. If you need additional rows, you may add them to the space just below the table. Quantity to Be Measured Symbol for Quantity Equipment for Measurement Describe the overall procedure to be used, referring to the table. Provide enough detail so that another student could replicate the experiment. including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table and/or include a simple diagram of the setup. The students' teacher has a video camera, which is not available for use in the experimental procedure described in part (b). The teacher uses the camera and video analysis software to generate the data shown in the table below. The table shows the sphere's horizontal and vertical displacements x and y. as measured from the location of the sphere's launch point. Horizontal Displacement :r: (m) Vertical Displacement y (m) (c) The students want to use the teacher's data to determine whether the claim that y is proportional to 2:2 is true. 1'. Indicate below which quantities could be graphed to determine whether the claim is true. You may use the remaining columns in the table, as needed, to record any quantities (including units) that are not already in the table. Vertical axisz Horizontal axis: ii. On the grid below. plot the appropriate quantities to determine whether the claim is true. Clearly scale and label all axes, including units as appropriate. iii. Do the data in the above graph support the claim that y is proportional to 3:2 is true? Yes _ No State your reasoning. (d) The graphs show the sphere's horizontal and vertical velocity components, oz and vy, as functions of time t, where t = 0 at the instant the sphere is launched. 0n the same graphs, sketch what v; and y would look like for a different launcher that has a launch speed 1% than :10. Vx Vy Launcher (Vertically Movable) 1 of Sphere Meterstick Plain Stand Carbon Paper Paper Stopwatch Experimental Apparatus The students have access to the experimental apparatus shown in the figure. Besides the sphere, the apparatus consists of the following. A launcher that launches the sphere horizontally, always with the same unknown speed vo A stand that can be used to adjust the height of the launcher above the floor The students also have several sheets of plain paper, several sheets of carbon paper, a meterstick, and a stopwatch. Whenever the sphere bounces off of the carbon paper, the carbon paper produces a visible mark on the sheet of plain paper that is under the carbon paper, as shown in the figure. Mark on Paper Marking the Paper with the Sphere K