AP Physics 1 2015 Free Response Question 4

Question: 
Two identical spheres emerge from an electronic device at the same time (t = 0) and from the same height (h = H), as shown below. Sphere A has no initial velocity and moves downward in a straight line. Sphere B moves with an initial horizontal velocity (v₀) and covers a horizontal distance D when it hits the ground. The two spheres reach the ground at the same time (t_f) though sphere B moves parabolically and travels a longer distance before landing. (Assume there is no air resistance.)

Students have to explain why the two identical spheres reach the ground at the same time if they are dropped simultaneously.

Scoring Guidelines: 

Indicate the horizontal motions of the two spheres do not affect their vertical motions.	1 point
Indicate the two spheres move with the same vertical velocity.	1 point
Indicate the two spheres move with the same vertical acceleration.	1 point
Indicate the two spheres falling from the same height would take the same time.	1 point
There is/are no incorrect or irrelevant statement(s).	1 point

(Source: http://apcentral.collegeboard.com/home)

Comments:
The purpose of this question is to assess students’ understanding of two-dimensional motion, as well as equations behind one sphere that is dropped from rest and a second identical sphere that is projected with an initial horizontal speed from the same height. In this post, we discuss a part of the question which is about why a sphere that is falling vertically in a straight line can reach the ground at the same time as the other sphere that is falling in a parabolic path. Importantly, the explanation of this phenomenon could include theoretical and empirical knowledge.

1. Theoretical idealization: Based on the scoring guidelines, a point can be awarded if students indicate that the two spheres falling the same height, would take the same time. That is, the horizontal velocity of the sphere does not affect its vertical velocity. In other words, the horizontal motion and vertical motion are independent of each other. However, we have assumed that the falling of an object is independent of its mass. Strictly speaking, mass independence is not valid for an observer on the ground (Lehavi & Galili, 2009). By using Newton’s law of gravitation or Newton’s third law, the Earth can accelerate toward the falling object. Thus, the statement that “all objects fall to the ground with the same acceleration” is a theoretical idealization and it is only approximately correct.

2. Empirical knowledge: Based on the scoring guidelines, another point can be awarded if students indicate that the difference in horizontal motion does not affect the vertical motion of the spheres. However, we should not simply deduce the independence of horizontal and vertical motions to be a theory. Similarly, we should not assume that two objects of different mass will reach the ground at the same time by simply using a thought experiment. Importantly, the falling of two spheres to the ground can be verified by experiments. Note that the air resistance acting on the two spheres are in different directions while they are falling toward the ground. Furthermore, it is empirically verified that the frictional drag on an object is proportional to the velocity or proportional to the square of the velocity depending on how fast the object is moving.

Nevertheless, if Einstein were a student taking this examination, he might pose the following thought experiment: “I stand at the window of a railway carriage which is traveling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the ‘positions’ traversed by the stone lie ‘in reality’ on a straight line or on a parabola? (Einstein, 1961, p. 10).” In essence, the so-called straight line or parabolic path is dependent on the velocity of the observer. Simply phrased, we can explain the object’s motion in terms of the observer’s frame of reference. Perhaps, based on the scoring guidelines, Einstein or some students could be penalized in answering this question by using the concept of inertial reference frame? 

Feynman’s insights and goofs?:
We can understand the question from the perspectives of empirical knowledge and theoretical idealization.
1. Empirical knowledge: Feynman explains that “[a] falling body moves horizontally without any change in horizontal motion, while it moves vertically the same way as it would move if the horizontal motion were zero. In other words, motions in the x-, y-, and z-directions are independent if the forces are not connected. (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).” Essentially, the motions of an object can be resolved into perpendicular directions that are independent of each other. Furthermore, in the Fig. 7-3 of Volume I of The Feynman Lectures on Physics, there is an apparatus that demonstrate the independence of vertical and horizontal motions (Feynman et al., 1963). Thus, the independence of the motions of the object in the x-, y-, and z-directions can also be considered as an empirical fact.

2. Theoretical idealization: Feynman writes that “[w]hat happens if we shoot a bullet faster and faster? Do not forget that the earth’s surface is curved. If we shoot it fast enough, then when it falls 16 feet it may be at just the same height above the ground as it was before. How can that be? It still falls, but the earth curves away, so it falls ‘around’ the earth (Feynman et al., 1963, section 7–4 Newton’s law of gravitation).” In other words, we have idealized the ground to be flat if the object falls within a short distance. This approximation does not hold if the object moves at a relatively high horizontal speed. In short, there are theoretical idealization and approximations involved in the falling of the object. The earth is not perfectly flat.

Note:

1. If Galileo were a student taking this examination, he might explain that the two spheres reach the ground at the same time because we have assumed that the air resistance is negligible. For example, in the Fourth Day of Dialogues Concerning Two New Sciences, Galileo (1638) writes that “if we consider only the resistance which the air offers to the motions studied by us, we shall see that it disturbs them all and disturbs them in an infinite variety of ways corresponding to the infinite variety in the form, weight, and velocity of the projectiles (p. 252).”

2. For another discussion of this question, you can visit the following website:
https://www.youtube.com/watch?v=x7UHl3Umeeg

References: 

1. Einstein, A. (1961/1916). Relativity: The Special and The General Theory. New York: Random House.

2. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Galileo, G. (1638/1954). Dialogues Concerning Two New Sciences (trans. H. Crew & A. de Salvio). New York: Dover.

4. Lehavi, Y., & Galili, I. (2009). The status of Galileo’s law of free-fall and its implications for physics education. American Journal of Physics, 77(5), 417-423.