An undergraduate student working with me this past year focused his thesis research on investigating student difficulties with projectile motion. The research consisted mostly of analyzing student responses to written problems, multiple-choice questions, and some clinical interviews. He focused mostly on student difficulties amid problem-solving, but also some questions targeting their reasoning about vectors not in the context of problem-solving.

In this post, I’m mostly just going to focus on the common difficulties that were observed in students’ problem-solving. Here are the five most common mistakes that showed up in our sample. All of these were somewhat familiar to me as an instructor, but the prevalence of some were surprising. All and all, 50% of students made at least one of these errors.

(a) Identifying the final velocity of the projectile as zero (and to a lesser extent initial velocity)

This was definitely the most common difficulty with upwards of 20% of students making this mistake. Students who made this error were very likely to make at least one of the other errors below.

Although it’s tempting to think that this error can merely be addressed by focusing students attention to the fact that we are talking about the speed before impact, some of our conversations with students suggest that it goes deeper. For some students, it seems that it connects with difficulties with instantaneous velocity. For example, I talked with a student who suggested the velocity just before impact must be zero because velocity is distance over time and there is no more distance to travel. Beyond difficulties with velocity, my guess would be this difficulty cannot be fully resolved without Newton’s Laws, whereby students are given explicit practice drawing free-body diagrams along various snapshots during the initiating launch, various points during free-fall, and during the impact stage. Our students do projectile motion before Newton’s laws and I think that’s a mistake.

(b) Identifying a non-zero acceleration in the x-direction (or identifying it as unknown)

The most common way this was instantiated was for students to identify both the acceleration in the x (and y) direction as 9.8 m/s/s. Some students, however, would indicate that the horizontal acceleration was an unknown to be solved for by placing a question mark next to it. Making this mistake seems to suggest students are not understanding the basic idea behind projectile motion.

(c) Difficulties translating written description of initial and final positions into x-y coordinates.

Much of this involved switching what would be correct forĀ x and y. For example, the problem might say that a golfer hits the ball from 10 meters above the green. Students would indicate that the 10 m was associated with the x-variable rather than the y-variable. There was a decent variety in exactly how this mistake was made.

A different student, working over this summer and fall, is doing some research to investigate the extent to which this difficulty stems from reading comprehension difficulties vs. coordinate system difficulties. One of the things we are asking students to do is to indicate all the places where x=0 on both axes that represent x-y coordinates and axes that represent x vs. t graphs.

(d) Finding launching or impact angles using triangles with distance information (rather than velocity information)

A correct way (and the way students are taught) to find the launch angle is by using trigonometry with a triangle composed from the initial velocity components. We observed lots of students solving for an angle using the distances. Basically, students end up solving for the angle describing the line that connects the initial and final positions rather than the launch angle.

We are looking into student understanding of the difference between these two angles in non-computational settings. It’s possible that students are actually confused about the two, or that during problem-solving that are just following a algorithm they don’t understand and aren’t really thinking about it all that much. My guess is this really stems from our lack of any instructional focus on kinematic vector concepts and its relationship to trajectory.

(e) Not clearly discriminating among velocity and velocity components.

Most commonly this would be observed where students would solve for the component of velocity, and then later be asked about the initial speed. Many students would identifying one or the other of the components as the speed, rather than combining them. A second way we observed this was when students might use x-component of velocity in a y-component kinematic equation or vice-versa. A final way this can arise is from students never finding components and using the initial speed in both x- and y-component equations. While some of this could certainly stem from carelessness, I’d bet most is related to vector issues.

A lot of these difficulties seem to relate to (i) difficulties with coordinate systems, (ii) not having a developed understanding of the vector nature of motion in 2D, and (iii) not having sufficient understanding of the fundamental idea (and even phenomenology) concerning the horizontal motion.