2023 AP Physics 2: Algebra-Based Free-Response Questions

Directions: Questions 1 and 4 are short free-response questions that require about 20 minutes each to answer and are worth 10 points each. Questions 2 and 3 are long free-response questions that require about 25 minutes each to answer and are worth 12 points each. Show your work for each part in the space provided after that part. 1. (10 points, suggested time 20 minutes) A rectangular tank with a mirrored bottom is filled with water (index of refraction n_w). A beam of light passes from air (index of refraction n_a) into the water at angle θ_i from the normal, as shown in Figure 1. Index of refraction n_w is greater than index of refraction n_a. (a) On the following diagram, sketch the entire path of the beam as the beam enters, travels through, and then exits the water. Figure 2 Sugar is then added to the water, resulting in a mixture that has a different index of refraction than water. A student considers two models, Model A and Model B, for how the sugar mixes with water. The models are shown in Figure 2. Model A: The sugar is uniformly mixed throughout the water, resulting in a mixture with index of refraction n_m such that n_m > n_w. Model B: Layers are formed of varying concentrations of sugar in the water. There are three distinct layers of equal volume. The top layer is only water (index of refraction n_w). The middle layer has the same concentration of sugar as the mixture in Model A (index of refraction n_m). The bottom layer has the highest concentration of sugar (index of refraction n_b). (b) Consider Model A. Briefly describe how the observed wavelength of light changes, if at all, as the beam travels from air into the mixture. (c) Relevant angles between the beam and the normal for the various layers present in models A and B are defined in the following table. Model A | Model B θ_i | Incident angle of the beam in air | θ_i | Incident angle of the beam in air θ_1 | Angle the beam makes with the normal in the mixture in Model A | θ_2 | Angle the beam makes with the normal in the top layer in Model B | | θ_3 | Angle the beam makes with the normal in the middle layer in Model B | | θ_4 | Angle the beam makes with the normal in the bottom layer in Model B i. Determine an expression for θ_4 in terms of θ_i, n_a, and n_b. ii. Rank the angles from greatest to least, with 1 being greatest. If two angles are the same value, give them the same ranking. ____ θ_1 ____ θ_2 ____ θ_3 ____ θ_4 Briefly explain your reasoning using appropriate physics principles and/or mathematical models. For the original tank filled with water, the beam is observed to exit the surface of the water a horizontal distance d_w from the entry point. For models A and B, the horizontal distances are d_A and d_B, respectively. (d) Determine whether d_A and d_B are each greater than, less than, or equal to d_w. It is NOT necessary to compare d_A to d_B. Briefly justify your answer.