2023 AP Physics C: Mechanics Free-Response Questions

Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this booklet in the spaces provided after each part. 1. A block sitting on a horizontal surface is pushed against a spring, Spring P, that is attached to a wall, compressing the spring a distance x, as shown in Figure 1. The block is then released from rest. The block slides along the horizontal surface and up a ramp, reaching a maximum height h_max,P. Frictional forces between the block and all surfaces are negligible. A student compares h_max,P for Spring P with the maximum height h_max,Q achieved with a different spring, Spring Q. Each spring exerts a force of magnitude F on the block that varies as a function of the distance x that the spring is compressed, as shown in Figure 2. For Spring P, F_P(x) = kx, where k = 100.0 N/m, and for Spring Q, F_Q(x) = Cx^{1/2}, where C = 20.0 N/m^{1/2}. (a) For the experiment, the block is pushed against one of the springs, compressing the spring a distance x = 0.010 m. The block is then released from rest. In Trial 1, Spring P is used, and in Trial 2, Spring Q is used. On the following dots, representing the block in Trials 1 and 2, draw and label the forces (not components) that are exerted on the block at the instant the block is released. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. The lengths of the horizontal vectors should represent the relative magnitude of the horizontal forces and the lengths of the vertical vectors should represent the relative magnitude of the vertical forces. Trial 1 (Spring P) Trial 2 (Spring Q) (b) i. What feature(s) of the graph in Figure 2 could be used to estimate the work done on the block by each spring as each spring is compressed? ii. There is one compression distance x_0 for which the maximum height h_max reached by the block is the same regardless of which spring, Spring P or Spring Q, is used. Predict whether the value of x_0 is greater than, less than, or equal to 0.040 m. Use the graph in Figure 2 to justify your answer. iii. On the axes provided, for both Spring P and Spring Q, sketch a graph of the maximum height h_max reached by the block as a function of the distance each spring is compressed for values of x ranging from 0 to 0.100 m. Clearly label the curve for Spring P and Spring Q. (c) Spring Q is attached to the wall and is at equilibrium, as shown in Figure 3. Block A has a mass of 0.120 kg and Block B has a mass of 0.070 kg. Block A is held at rest at the top of the ramp and Block B is at rest on the horizontal surface. The student releases Block A, and it moves down the ramp and collides with Block B. The change in vertical height of Block A is H = 0.75 m. After the collision, the blocks stick together and move to the right, compressing the spring. Frictional forces between the blocks and all surfaces are negligible. i. Calculate the velocity of the two-block system immediately after the collision between Blocks A and B. ii. Calculate the maximum compression of the spring. (d) Spring Q where F_Q(x) = Cx^{1/2} is replaced by a different nonlinear spring, Spring R, and the procedure described in part (c) is repeated. For Spring R, F_R(x) = Dx^{1/2}. The maximum compression of Spring R is greater than the maximum compression of Spring Q. Which of the following correctly compares the constants C and D? ___ C < D ___ C > D ___ C = D Briefly justify your answer. © 2023 College Board. Visit College Board on the web: collegeboard.org.