Question

1. Teresa Drops a Stone. (Allotted Time - 40 minutes) Teresa drops a stone from rest from the top of an extremely tall building on a huge planet with essentially no atmosphere. You may assume that the stone experiences a constant gravitational acceleration as it falls to the planet’s surface, but the magnitude of this acceleration may not be equal to that on Earth. Let d1 be the distance the rock travels during the 1st second of its fall, let d2 be the distance the rock travels during the 2nd second of its fall, let d3 be the distance the rock travels during the 3rd second of its fall, and so on. In general for any positive integer n,dn is the distance the stone travels during the nth second of its fall.

1.1. Based on physical/intuitive arguments and without using any math, state whether you expect d1 to be greater than, less than, or equal to d2, and explain your conclusion.

1.2. Suppose that the acceleration due to gravity near the surface of the planet has magnitude 10g . Com- pute numerical values for d1, d2, and the ratio d1/d2.

1.3. If the acceleration due to gravity near the surface of the planet were to have a smaller magnitude, which of the quantities d1, d2, and the ratio d1/d2 would change, and which if any would not? For any of these quantities that would change, would it get smaller, larger, or stay the same. For any one of them that wouldn’t, justify why you believe that it wouldn’t.

1.4. Imagine that Teresa does this stone dropping experiment on two different planets, planet A and planetB having accelerations due to gravity gA and gB = 3gA respectively. Suppose that the surface of the planets is taken to be y = 0 with positive y pointing upward away from the surfaces, and the stones are dropped at rest from the same height on both planets. Sketch the velocity versus time graphs for the stones on the two planets on the same set of axes so that they can easily be compared. Let t = 0 correspond to when the stones are dropped, and include up to the time when they impact the ground. Sketch also the position versus time graphs on another set of axes so that they can easily be compared. Compare and contrast the graphs, and comment on what they reveal about which stone hits its planet’s surface first. How does the the total area under the velocity versus time graphs compare from t = 0 to when the stones hit the ground?